Models.nsm.Temperature Dependent Plasticity

8/10/2019 Models.nsm.Temperature Dependent Plasticity

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Solved with COMSOL Multiphysics 4.3a

2 0 1 2 C O M S O L 1 | TE M P E R A T U R E - D E P E N D E N T P L A S T I C I T Y I N P R E S S U R E V E S S E L

Tempera ture-Dependent P las t ic i ty inPressure Vesse l

Introduction

This example demonstrates how to use temperature dependent materials within theNonlinear Structural Materials Module. Material data such as Youngs modulus, yield

stress and strain hardening have strong temperature dependencies.

A large container holds pressurized hot water. Several pipes are attached to the pressure vessel. Those pipes can rapidly transfer cold water in case of an emergency cooling. Thepressure vessel is made of carbon steel with an internal cladding of stainless steel. Incase of a fast temperature transient, the differences in thermal expansion propertiesbetween the materials will cause high stresses.

Model Definition

G E O M E T R Y

The pressure vessel has the shape of a closed cylinder. Four pipes are attached at twolevels along its height. At each level, the pipes are equidistantly spaced around thecontainer.


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The pipes are welded into the vessel and the welding can be considered as a chamferbetween those two parts. The structure, together with its key dimensions, is presented

in Figure 1 .

pipe

vessel vessel

pipe

l v

lp

tp

r p

t vr v

Figure 1: Pressure vessel and the dimensions for the vessel-pipe connection.

The structure has the following dimensions:

Inner vesse l radius, r v = 1000 mm,

Inner pipe radius, r p = 60 mm,

Vessel thickness, t v = 100 mm,

Pipe thickness, tp = 40 mm.

The vessel length, l v, and the pipe length, lp, are large compared to the thickness ofboth parts. For modeling purposes, they need to be large enough so that local effectsat the vessel-pipe connection can be disregarded. The chamfer extends 20 mm fromthe co rner at the connection between the pipe and the vessel.

The dual material consists of a thin 10 mm layer of stainless steel (dark gray inFigure 1 ) that faces the water, and carbon steel (light gray in Figure 1 ) that faces the

outside air.In order to save computational time, only the connection between one pipe and the

vessel is modeled, as shown on the left image of Figure 1 .


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M AT E R I A L M O D E L

The thermoelastic material data of stainless steel is given in Table 1 . Table 2 shows the

yield stress as a function of plastic strain at temperatures of 20, 100, 200 and 300 C.

Temperature ( C) 20 100 200 300

E (GPa) 194 189 179 175 (1/ C) 1610 -6 16.510-6 1710-6 17.510-6

cp (J/(kgK)) 482 498 515 524

k (W/(mK)) 13.9 14.9 17.0 18.0

Temperature ( C) 20 100 200 300

ys (0.0) (MPa) 228 190 156 140 ys (0.0004) (MPa) 232 195 160 144 ys (0.001) (MPa) 238 201 166 148

ys (0.002) (MPa) 246 210 173 155 ys (0.004) (MPa) 250 215 177 158 ys (0.001) (MPa) 263 230 189 169

The yield stress of the carbon steel is two or three times higher than that of stainlesssteel. It is therefore considered as elastic. Its material properties are shown in Table 3 .

Temperature ( C) 20 100 200 300E (GPa) 208 202 196 189 (1/ C) 10.910 -6 12.410-6 13.810-6 14.910-6

cp (J/(kgK)) 489 519 546 569

k (W/(mK)) 51.2 48.3 45.5 42.7

The heat transfer coefficient between steel and air is 10 W/(m 2K), and between steeland water is 100 W/(m 2K).

B O U N D A R Y C O N D I T I O N S

A pressure of 70 bar acts on the vessel and pipes inner walls. The temperature on theinside of the pressure vessel is initially at 280 C, while the outside air remains at 50 C.Suddenly and instantaneously, cold water at 20 C is pumped through the pipe into the

vessel, where the hot water needs 30 minutes to cool down to 20 C. The coolingspeed is constant.

TABLE 1: THERMOELASTIC MATERIAL DATA OF STAINLESS STEEL.

TABLE 2: TEMPERATURE DEPENDENT YIELD STRESS OF STAINLESS STEEL

TABLE 3: THERMOELASTIC MATERIAL DATA OF CARBON STEEL


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M O D E L A S S U M P T I O N S

Due to symmetries only a 45 sector of the vessel is modeled. The influence of the hot

water pressure at the end of the vessel is approximated with an axial stress of 33.3 MPa, which is 4.76 times the inner pressure. The parameters l v and lp are both set to200 mm.

Results and Discussion

Two studies are performed in this analysis. In an initial step, the mechanical and

thermal stationary state is computed. This serves as initial conditions for a transientstep, where the cold water flows through the pipe, cooling the initially hot water in the vessel. A comparison of the temperature profiles, before and after the event, are shownin Figure 2 .

Figure 2: Temperature profiles before and after the cooling.

After 30 min, the water inside the vessel has cooled down to 20 C, but the containeris still locally more than 100 C hotter. There are large gradients in the thermal strains.


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The development of plastic strains in the stainless steel layer is demonstrated inFigure 3 .

Figure 3: Plasticity before and after the cooling.

From Figure 3 , it can be seen that a plastic zone develops as the vessel cools down.Initially, when the vessel is at steady state, some plastic strain is generated by stressescaused by differences in the thermal expansion of the two steels. In a real structure,

such stresses would have been relaxed after the first service cycle. In the transient study, when the vessel cools down and the yield limit increases, the pipe deforms plasticallyin other locations.


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As the temperature decreases, the yield stress increases, so hot parts are more sensitiveto high stresses. Figure 4 shows the effective stress after 30 min of cooling.

Figure 4: Effective stress after cooling down the hot water.

Notes About the COMSOL Implementation

COMSOL Multiphysics can handle material data depending on several parameters.Use Interpolation functions, which are selectable form the Definitions or Materialsnodes. You can either type in the data in a table or define your function in a text file.

You can use the symbol % in a text file to include comments or headers.

Interpolation functions can handle any number of arguments. For convenience, specifyunits (Pa, m, s, and so on) for the function and arguments. When applicable,

COMSOL Multiphysics will automatically scale any input into the correct unit. Formore details see the sections Functions in the COMSOL Multiphysics ReferenceGuide and Using Functions in the COMSOL Multiphysics Users Guide .

In this model there are two cuts which are almost symmetry cuts in the sense thatthey should stay plane, but are still allowed to move in the normal direction. One isthe cut in the pipe, and the other the cut through the pressure vessel wall. In order toaccomplish this, the displacement in the normal direction is prescribed for the whole
http://../mph/mph_ref_general.pdfhttp://../mph/mph_ug_material.pdfhttp://../mph/mph_ug_material.pdfhttp://../mph/mph_ref_general.pdf


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boundary, but not to a constant value. It is prescribed to be the displacement in onepoint on the boundary. Thus it is constant, but yet determined by the solution.

Model Library path: Nonlinear_Structural_Materials_Module/Plasticity/temperature_dependent_plasticity

Modeling Instructions

M O D E L W I Z A R D

1 Go to the Model Wizard window.

2 Click Next .

3 In the Add physics tree, select Structural Mechanics>Thermal Stress (ts) .

4 Click Next .

5 Find the Studies subsection. In the tree, select Preset Studies>Stationary .6 Click Finish .

G L O B A L D E F I N I T I O N S

Parameters1 In the Model Builder window, right-click Global Definitions and choose Parameters .

2 In the Parameters settings window, locate the Parameters section.

3 In the table, enter the following settings:

The thermal shock caused by the cold water is time dependent. Therefore the variable for time, t , needs to be set to zero so that the heat boundary conditions canbe evaluated also in the static analysis.

If you do not want to build all the geometry, you can load the geometry sequence fromthe stored model. In the Model Builder window, under Model 1 right-click Geometry 1and choose Insert Sequence from File . Browse to the models Model Library folder and

Name Expression Description

internalPressure 70[bar] Internal pressure

t 0[s] Time variable; used forstationary analysis


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double-click the file temperature_dependent_plasticity.mph . You can thencontinue to the Definitions section on page 12 .

To build the geometry from scratch, continue here.

G E O M E T R Y 1

Work Plane 11 In the Model Builder window, under Model 1 right-click Geometry 1 and choose Work

Plane .

2 In the Work Plane settings window, locate the Work Plane section.3 From the Plane list, choose xz-plane .

Bzier Polygon 11 In the Model Builder window, under Model 1>Geometry 1>Work Plane 1 right-click

Plane Geometry and choose Bzier Polygon .

2 In the Bzier Polygon settings window, locate the Polygon Segments section.

3 Find the Added segments subsection. Click the Add Linear button.4 Find the Control points subsection. In row 1, set xw to 1.3 and yw to 0.07 .

5 In row 2, set xw to 1.08 and yw to 0.07 .

6 Find the Added segments subsection. Click the Add Linear button.

7 Find the Control points subsection. In row 2, set yw to 0.14 .


9 Find the Control points subsection. In row 2, set xw to 1.12 and yw to 0.1 .10 Find the Added segments subsection. Click the Add Linear button.

11 Find the Control points subsection. In row 2, set xw to 1.3 and yw to 0.1 .


13 Find the Control points subsection. Click the Close Curve button.

14 Click the Build Selected button.

Revolve 11 In the Model Builder window, under Model 1>Geometry 1 right-click Work Plane 1 and

choose Revolve .

2 In the Revolve settings window, locate the Revolution Axis section.

3 Find the Direction of revolution axis subsection. In the xw edit field, type 1 .

4 In the yw edit field, type 0 .


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5 Locate the Revolution Angles section. In the End angle edit field, type -90 .


Work Plane 21 In the Model Builder window, right-click Geometry 1 and choose Work Plane .

2 In the Work Plane settings window, locate the Work Plane section.

3 From the Plane list, choose xz-plane .

Rectangle 11 In the Model Builder window, under Model 1>Geometry 1>Work Plane 2 right-click

Plane Geometry and choose Rectangle .

2 In the Rectangle settings window, locate the Size section.

3 In the Width edit field, type 0.09 .

4 In the Height edit field, type 0.3 .

5 Locate the Position section. In the xw edit field, type 1.01 .



choose Revolve .

2 In the Revolve settings window, locate the Revolution Angles section.

3 In the End angle edit field, type 45 .


5 Click the Zoom Extents button on the Graphics toolbar.

Cylinder 11 In the Model Builder window, right-click Geometry 1 and choose Cylinder .

2 In the Cylinder settings window, locate the Size and Shape section.

3 In the Radius edit field, type 0.07 .


5 Locate the Position section. In the x edit field, type 0.95 .

6 Locate the Axis section. From the Axis type list, choose Cartesian .

7 In the x edit field, type 1 .

8 In the z edit field, type 0 .



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Difference 11 Right-click Geometry 1 and choose Boolean Operations>Difference .

2 Select the objects rev2 and rev1 only.3 In the Difference settings window, locate the Difference section.

4 Under Objects to subtract , click Activate Selection .

5 Select the object cyl1 only.


Work Plane 31 Right-click Geometry 1 and choose Work Plane .

2 In the Work Plane settings window, locate the Work Plane section.

3 From the Plane list, choose xz-plane .




3 In the Width edit field, type 0.3 .


5 Locate the Position section. In the xw edit field, type 1 .

6 In the yw edit field, type 0.06 .


Work Plane 3In the Model Builder window, under Model 1>Geometry 1 right-click Work Plane 3 andchoose Revolve .

Revolve 31 In the Model Builder window, under Model 1>Geometry 1>Work Plane 3>Plane

Geometry right-click Rectangle 1 and choose Build Selected .

2 In the Model Builder window, under Model 1>Geometry 1 click Revolve 3 .3 In the Revolve settings window, locate the Revolution Axis section.

4 Find the Direction of revolution axis subsection. In the xw edit field, type 1 .

5 In the yw edit field, type 0 .

6 Locate the Revolution Angles section. In the End angle edit field, type -90 .



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Work Plane 41 In the Model Builder window, right-click Geometry 1 and choose Work Plane .

2 In the Work Plane settings window, locate the Work Plane section.3 From the Plane list, choose xz-plane .




3 In the Height edit field, type 1.0 .4 In the Width edit field, type 0.01 .


6 Locate the Position section. In the yw edit field, type 0 .

7 In the xw edit field, type 1 .



choose Revolve .

2 In the Revolve settings window, locate the Revolution Angles section.

3 In the End angle edit field, type 45 .


Cylinder 21 In the Model Builder window, right-click Geometry 1 and choose Cylinder .

2 In the Cylinder settings window, locate the Size and Shape section.

3 In the Radius edit field, type 0.06 .


5 Locate the Position section. In the x edit field, type 0.95 .

6 Locate the Axis section. From the Axis type list, choose Cartesian .

7 In the x edit field, type 1 .

8 In the z edit field, type 0 .



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M O D E L 1

In the Model Builder window, expand the Model 1 node.

G E O M E T R Y 1

In the Model Builder window, expand the Model 1>Geometry 1 node.

Difference 21 Right-click Geometry 1 and choose Boolean Operations>Difference .

2 Select the objects rev4 and dif1 only.

3 In the Difference settings window, locate the Difference section.

4 Under Objects to subtract , click Activate Selection .

5 Select the object cyl2 only.


Ignore Faces 11 Right-click Geometry 1 and choose Virtual Operations>Ignore Faces .

2 On the object fin, select Boundaries 11, 15, 17, 23, 26, and 28 only.3 Click the Build Selected button.

Ignore Edges 11 Right-click Geometry 1 and choose Virtual Operations>Ignore Edges .

2 On the object igf1, select Edge 21 only.


Load the table containing yield stress as function of plastic strain and temperature.

D E F I N I T I O N S

Interpolation 11 In the Model Builder window, under Model 1 right-click Definitions and choose

Functions>Interpolation .

2 In the Interpolation settings window, locate the Parameters section.

3 From the Data source list, choose File.

4 Click the Browse button.

5 Browse to the models Model Library folder and double-click the filetemperature_dependent_plasticity_function.txt .


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6 Find the Functions subsection. In the table, enter the following settings:

7 Locate the Units section. In the Arguments edit field, type degC,1 .

8 In the Function edit field, type Pa .

9 Locate the Parameters section. Click the Import button.

T H E R M A L S T R E S S

Thermal Linear Elastic Material 21 In the Model Builder window, under Model 1 right-click Thermal Stress and choose

Thermal Linear Elastic Material .

2 In the Model Builder window, expand the Thermal Stress node, then click ThermalLinear Elastic Material 2 .

3 Select Domain 2 only.

Plasticity 11 In the Model Builder window, under Model 1>Thermal Stress right-click Thermal

Linear Elastic Material 1 and choose Plasticity .

2 In the Model Builder window, expand the Thermal Linear Elastic Material 1 node, thenclick Plasticity 1 .

3 In the Plasticity settings window, locate the Plasticity Model section.

4 From the Isotropic hardening list, choose Use hardening function data .5 In the h( pe ) edit field, type sY(T,ts.epeGp)-sY(T,0) .

The hardening function is the stress increase from the initial yield stress. As the fullstress-strain curve is given, subtract the stress at zero strain.

M AT E R I A L S

Material 11 In the Model Builder window, under Model 1 right-click Materials and choose Material .

2 Right-click Material 1 and choose Rename .

3 Go to the Rename Material dialog box and type Stainless steel in the New name edit field.

4 Click OK.


Function namesY


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5 Locate the Units section. In the Arguments edit field, type degC .

6 In the Function edit field, type 1/K .

Stainless steel Add thermal conductivity as function of temperature for the stainless steel.

Interpolation 31 Right-click Basic and choose Functions>Interpolation .


3In the

Function name edit field, type

fK.



6 In the Function edit field, type W/(m*degC) .

Stainless steel Add heat capacity as function of temperature for the stainless steel.

Interpolation 41 Right-click Basic and choose Functions>Interpolation .2 In the Interpolation settings window, locate the Parameters section.

3 In the Function name edit field, type fCp .

t f(t)

20 16e-6

100 16.5e-6

200 17e-6

300 17.5e-6

t f(t)

20 13.9

100 15.5

200 16.8

300 17.8


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6 In the Function edit field, type J/(kg*degC) .

Stainless steel 1 In the Model Builder window, click Stainless steel .

2 In the Material settings window, locate the Material Contents section.


Material 21 In the Model Builder window, right-click Materials and choose Material .

2 Right-click Material 2 and choose Rename .3 Go to the Rename Material dialog box and type Carbon steel in the New name edit

field.

4 Click OK.


Carbon steel

1 In the Material settings window, locate the Material Contents section.2 In the table, enter the following settings:

t f(t)

20 482

100 504

200 521

300 530

Property Name Value

Young's modulus E fE(T)Coefficient of thermal expansion alpha fA(T)

Thermal conductivity k fK(T)

Heat capacity at constant pressure Cp fCp(T)

Property Name Value

Poisson's ratio nu 0.3

Density rho 8000


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Add Young's modulus as function of temperature for the carbon steel.

Interpolation 11 In the Model Builder window, under Model 1>Materials>Carbon steel right-click Basic

and choose Functions>Interpolation .


3 In the Function name edit field, type fE .



6 In the Function edit field, type GPa .

Carbon steel Add thermal expansion as function of temperature for the carbon steel.



3 In the Function name edit field, type fA .



6 In the Function edit field, type 1/K .

Carbon steel Add thermal conductivity as function of temperature for the carbon steel.

t f(t)

20 208

100 202

200 196

300 189

t f(t)

20 10.9e-6

100 12.4e-6

200 13.8e-6

300 14.9e-6


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2 In the Interpolation settings window, locate the Parameters section.3 In the Function name edit field, type fK .



6 In the Function edit field, type W/(m*degC) .

Carbon steel Add heat capacity as function of temperature for the carbon steel.



3 In the Function name edit field, type fCp .



6 In the Function edit field, type J/(kg*degC) .Carbon steel 1 In the Model Builder window, under Model 1>Materials click Carbon steel .

2 In the Material settings window, locate the Material Contents section.

t f(t)

20 51.2

100 48.3

200 45.5

300 42.7

t f(t)20 489

100 519

200 546

300 569


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D E F I N I T I O N S

Add the time history for the temperature of the water in the pipe.Step 11 In the Model Builder window, under Model 1 right-click Definitions and choose

Functions>Step .

2 In the Model Builder window, expand the Definitions node, then click Step 1 .

3 In the Step settings window, locate the Function Name section.

4 In the Function name edit field, type pipeWaterTemp .5 Locate the Parameters section. In the Location edit field, type 1 .

6 In the From edit field, type 280 .

7 In the To edit field, type 20 .

Add the time history for the temperature of the water in the pressure vessel.

Interpolation 21 In the Model Builder window, right-click Definitions and choose

Functions>Interpolation .


3 In the Function name edit field, type vesselWaterTemp .


5 Locate the Units section. In the Arguments edit field, type s .

6 In the Function edit field, type degC .

Add an operator for capturing the x-displacement in a point at the pipe end.

Property Name Value

Young's modulus E fE(T)

Coefficient of thermal expansion alpha fA(T)

Thermal conductivity k fK(T)

Heat capacity at constant pressure Cp fCp(T)

t f(t)

0 280

1800 20


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Boundary Load 11 Right-click Thermal Stress and choose the boundary condition Solid

Mechanics>Boundary Load .2 Select Boundaries 2 and 10 only.

3 In the Boundary Load settings window, locate the Force section.

4 From the Load type list, choose Pressure .

5 In the p edit field, type internalPressure .

Boundary Load 2

1 Right-click Thermal Stress and choose the boundary condition SolidMechanics>Boundary Load .

2 Select Boundaries 4 and 8 only.

3 In the Boundary Load settings window, locate the Force section.

4 In the F A table, enter the following settings:

Heat Flux 11 Right-click Thermal Stress and choose the boundary condition Heat Transfer>Heat

Flux.

2 Select Boundaries 9, 14, and 15 only.

3 In the Heat Flux settings window, locate the Heat Flux section.

4 Click the Inward heat flux button.

5 In the h edit field, type 10 .

6 In the T ext edit field, type 50[degC] .


Flux.2 In the Heat Flux settings window, locate the Heat Flux section.


4 Select Boundary 10 only.


6 In the T ext edit field, type pipeWaterTemp(t[1/s])[degC] .

0 x

0 y

4.76*internalPressure z


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Flux.2 In the Heat Flux settings window, locate the Heat Flux section.


4 Select Boundary 2 only.


6 In the T ext edit field, type vesselWaterTemp(t) .

Make the cuts stay plane under deformation by prescribing the normal displacementover the entire boundary to be equal to the value in a single point in the boundary.

Prescribed Displacement 11 Right-click Thermal Stress and choose the boundary condition Solid

Mechanics>Prescribed Displacement .


3 In the Prescribed Displacement settings window, locate the Boundary Selection section.

4 Click Clear Selection .


6 In the Prescribed Displacement settings window, locate the Prescribed Displacement section.

7 Select the Prescribed in x direction check box.

8 In the u 0 edit field, type edgeX .

Prescribed Displacement 21 Right-click Thermal Stress and choose the boundary condition Solid

Mechanics>Prescribed Displacement .


3 In the Prescribed Displacement settings window, locate the Prescribed Displacement section.

4 Select the Prescribed in z direction check box.

5 In the w0 edit field, type edgeZ .

l d h l h


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S T U D Y 1

Step 2: Time Dependent1 In the Model Builder window, right-click Study 1 and choose Study Steps>Time

Dependent .

2 In the Time Dependent settings window, locate the Study Settings section.

3 In the Times edit field, type range(0,60,1800) .

Analyze half an hour, storing the results once every minute.

Solver 11 Right-click Study 1 and choose Show Default Solver .2 In the Model Builder window, expand the Solver 1 node, then click Time-Dependent

Solver 1 .

3 In the Time-Dependent Solver settings window, click to expand the Time Stepping section.

Since placticity is a path dependent phenomenon, it is important to capture thedevelopment accurately, so you want to limit the time steps.

4 Select the Maximum step check box.

5 In the associated edit field, type 60 .

6 In the Model Builder window, right-click Study 1 and choose Compute .

R E S U LT S

Stress (ts)1 In the Model Builder window, expand the Results>Stress (ts)>Surface 1 node, then

click Deformation .

2 In the Deformation settings window, locate the Scale section.

3 Select the Scale factor check box.

4 In the associated edit field, type 0 .

5 Click the Plot button.

Stress (ts) 11 In the Model Builder window, right-click Stress (ts) and choose Duplicate .

2 Right-click Stress (ts) 1 and choose Rename .

3 Go to the Rename 3D Plot Group dialog box and type Plastic strain in the Newname edit field.

4 Click OK.


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Models.nsm.Temperature Dependent Plasticity

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