An elastic plate with a crack of length 2a in its center subjected to uniform longitudinal tensile stress at one end and clamped at the other end as shown in figure below. An FEM analysis of the 2-D elastic center-cracked tension plate is carried out and the value of the mode I (crack-opening mode) Stress Intensity Factor (SIFs) for the center-cracked plate is calculated.
Specimen geometry: Length of the plate l = 400 mm, and height h = 100 mm.
Material: mild steel having Young's modulus E = 210 GPa and Poisson's ratio v = 0.3.
Crack: A crack is placed perpendicular to the loading direction in the center of the plate and has a length of 20 mm. The center - cracked tension plate is assumed to be in the plane strain condition in the analysis.
Boundary conditions: The elastic plate is subjected to a uniform tensile stress in the longitudinal direction at the right end and clamped to a rigid wall at the left hand.
A quarter model of the center-cracked tension plate is created, since the plate is symmetric about the horizontal and vertical center lines.
We use the singular element or the quarter point element which can interpolate the stress distribution in the vicinity of the crack tip at which stress has the 1/(r^1/2) singularity where r is the distance from the crack tip (r/a <<1). An ordinary isoparametric element as "Quad 8node 82" has nodes at the corners and also at the midpoint on each side of the element. A singular element, however, has the midpoint moved one-quarter side distance from the original midpoint position to the node which is placed at the crack tip position. This is the reason why the singular element is often called the quarter point element instead. ANSYS software is equipped only with a 2-D triangular singular element, but neither with 2-D rectangular nor with 3-D singular elements. Around the node at the crack tip, a circular area is created and is created and is divided into a designated number of triangular singular elements. Each triangular singular element has its vertex placed at the crack tip position and has the quarter points on the two sides joining the vertex and the other two nodes.
In order to create the singular elements, the plate area must be created via keypoints set at the four corner points and at the crack tip position on the left-end side of the quarter plate area.
1). From Main Menu click Preprocessor - Modeling - Create - Keypoints - In Active CS. The Create Keypoints in Active Coordinates System window opens. Input each keypoints as (0,0,0), (200,0,0), (200,50,0), (0,50,0), and (0,10,0), respectively. Finally click OK to close the window.
2). Then create the plate area via the five points. From Main Menu click Preprocessor - Modeling - Create - Area - Arbitrary - Through KPs. The Create Area thru KPs window opens. Click Key points number 1 through 5 one by one counter-clockwise.
Click OK to close the window and create the area.
3). To define material properties from Main Menu click Preprocessor - Material Props - Material Models. The Define Material Model Behavior window opens. Click Structural - Linear - Elastic - Isotropic. Input the value of EX = 2.1e5 and PRXY = 0.3 and click OK.
Exit from the Define Material Model Behavior window by selecting Exit in the Material menu of the window.
4). To define the element, from Main Menu click Preprocessor - Element Type - Add/Edit/Delete. Element Types window appears. Click on Add button. Library of Element Types window opens. From left column select Structural Mass - Solid and from right column select Quad 8node 82. Click OK to close the window.
PLANE82 is added to the Element Types window.
In Element Types window click Options button to open the PLANE82 element type options window. Select the Plane strain item in the Element behavior box and click OK to return to the Element Types window. Click the Close button.
5). Sizing of the elements:
Before meshing, the crack tip point around which the triangular singular elements will be created must be specified. From Main Menu click Preprocessor - Meshing - Size Cntrls - Concentrate KPs - Create. The Concentration Keypoint window opens.
Display the key points in the ANSYS Graphic window. From Menu click Plot - Keypoints - Keypoints.
Pick key point number 5 and then click OK button.
Another Concentration Keypoint window opens. Enter NPT = 5, DELR = 2, RRAT = 0.5, NTHET = 6, and KCTIP = Skewed 1/4pt. Skewed 1/4 pt means that the mid nodes of the sides of the elements which contain key point 5 are the quarter points of the elements. Click OK button.
The size of the meshes other than the singular elements and the elements adjacent to them can be controlled by the same procedures as previous section.
From Main Menu click Preprocessor - Meshing - Size Cntrls - Manual Size - Global - Size. The Global Element Sizes window opens. Input 1.5 in the SIZE box and click OK button.
6). Meshing:
From Main Menu click Preprocessor - Meshing - Mesh - Areas - Free. The Mesh Areas window opens. Pick the area and click OK.
The warning window appears due to the existence of six singular elements. Click Close button and proceed to the next operation below.
Figure below shows the plate area meshed by ordinary 8-node isoparametric finite elements except for the vicinity of the crack tip where we have six singular elements.
Figure below is an enlarged view of the singular elements around the crack tip showing that six triangular elements are placed in a radial manner and that the size of the second row of elements is half the radius of the first row of elements, i,e., triangular singular elements.
7). Boundary Conditions:
Due to the symmetry, the constraint conditions of the quarter plate model are UX-fixed condition on the left end side, that is, the line between key points 4 and 5, and UY-fixed condition on the bottom side of the quarter plate model.
From Main Menu click Solution - Define Loads - Apply - Structural - Displacement - On Lines. The Apply U, ROT on Lines window opens. Confirming that the Pick and Single buttons are selected, move the upward arrow onto the line between key points 4 and 5 and click the left button of the mouse.
Click the OK button to display another Apply U, ROT on Lines window. Select UX and click OK.
Again from Main Menu click Solution - Define Loads - Apply - Structural - Displacement - On Lines. The Apply U, ROT on Lines window opens. Confirming that the Pick and Single buttons are selected, move the upward arrow onto the bottom line and click the left button of the mouse.
Click the OK button to display another Apply U, ROT on Lines window. Select UY and click OK.
8). To apply uniform longitudinal stress on the right end of the quarter plate model,
From Main Menu click Solution - Define Loads - Apply - Structural - Pressure - On Lines. Apply PRES on Lines window opens and the upward arrow appears when the mouse cursor is moved to the ANSYS Graphics window. Pick the right-end side of the quarter plate area and click OK.
Another Apply PRES on Lines window opens. Select Constant value in the Apply PRES on Lines as a box and input -10 (MPa) in the VALUE Load PRES value box and leave a blank in the value box. Click OK to define a uniform tensile stress of 10 MPa applied to the right end of the quarter plate model.
9). Solution Procedures:
From Main Menu click Solution - Solve - Current LS. The Solve Current Load Step and /STATUS Command windows appear.
Click OK button in the Solve Current Load Step window to begin the solution of the current load step. The Verify window opens . Proceed to the next operation below by clicking Yes button in the window.
When the solution is completed, the Note window appears. Click the Close button to close the Note window.
10). Contour Plot of Stress:
From Main Menu click General Postproc - Plot Results - Contour Plot - Nodal Solution. The Contour Nodal Solution Data window opens. Select Stress and X-Component of stress. Then select Deformed shape with deformed edge in the Undisplaced shape key box to compare the shapes of the cracked plate before and after deformation. Then click OK.
X-Component of stress:
Figure below is an enlarged view of the longitudinal stress distribution around the crack tip showing that very high tensile stress is induced at the crack tip whereas almost zero stress around the crack surface and that the crack shape is parabolic.
Material: mild steel having Young's modulus E = 210 GPa and Poisson's ratio v = 0.3.
Crack: A crack is placed perpendicular to the loading direction in the center of the plate and has a length of 20 mm. The center - cracked tension plate is assumed to be in the plane strain condition in the analysis.
Boundary conditions: The elastic plate is subjected to a uniform tensile stress in the longitudinal direction at the right end and clamped to a rigid wall at the left hand.
A quarter model of the center-cracked tension plate is created, since the plate is symmetric about the horizontal and vertical center lines.
We use the singular element or the quarter point element which can interpolate the stress distribution in the vicinity of the crack tip at which stress has the 1/(r^1/2) singularity where r is the distance from the crack tip (r/a <<1). An ordinary isoparametric element as "Quad 8node 82" has nodes at the corners and also at the midpoint on each side of the element. A singular element, however, has the midpoint moved one-quarter side distance from the original midpoint position to the node which is placed at the crack tip position. This is the reason why the singular element is often called the quarter point element instead. ANSYS software is equipped only with a 2-D triangular singular element, but neither with 2-D rectangular nor with 3-D singular elements. Around the node at the crack tip, a circular area is created and is created and is divided into a designated number of triangular singular elements. Each triangular singular element has its vertex placed at the crack tip position and has the quarter points on the two sides joining the vertex and the other two nodes.
In order to create the singular elements, the plate area must be created via keypoints set at the four corner points and at the crack tip position on the left-end side of the quarter plate area.
To facilitate
the modeling of two coincident faces, a very small opening of the crack needs to be created. A recommended geometry of the opening is shown
in figure below:
The crack-tip region is meshed using quarter-point (singular)
8-node quadrilateral elements (PLANE82).
2). Then create the plate area via the five points. From Main Menu click Preprocessor - Modeling - Create - Area - Arbitrary - Through KPs. The Create Area thru KPs window opens. Click Key points number 1 through 5 one by one counter-clockwise.
Click OK to close the window and create the area.
3). To define material properties from Main Menu click Preprocessor - Material Props - Material Models. The Define Material Model Behavior window opens. Click Structural - Linear - Elastic - Isotropic. Input the value of EX = 2.1e5 and PRXY = 0.3 and click OK.
Exit from the Define Material Model Behavior window by selecting Exit in the Material menu of the window.
4). To define the element, from Main Menu click Preprocessor - Element Type - Add/Edit/Delete. Element Types window appears. Click on Add button. Library of Element Types window opens. From left column select Structural Mass - Solid and from right column select Quad 8node 82. Click OK to close the window.
PLANE82 is added to the Element Types window.
In Element Types window click Options button to open the PLANE82 element type options window. Select the Plane strain item in the Element behavior box and click OK to return to the Element Types window. Click the Close button.
5). Sizing of the elements:
Before meshing, the crack tip point around which the triangular singular elements will be created must be specified. From Main Menu click Preprocessor - Meshing - Size Cntrls - Concentrate KPs - Create. The Concentration Keypoint window opens.
Display the key points in the ANSYS Graphic window. From Menu click Plot - Keypoints - Keypoints.
Pick key point number 5 and then click OK button.
Another Concentration Keypoint window opens. Enter NPT = 5, DELR = 2, RRAT = 0.5, NTHET = 6, and KCTIP = Skewed 1/4pt. Skewed 1/4 pt means that the mid nodes of the sides of the elements which contain key point 5 are the quarter points of the elements. Click OK button.
The size of the meshes other than the singular elements and the elements adjacent to them can be controlled by the same procedures as previous section.
From Main Menu click Preprocessor - Meshing - Size Cntrls - Manual Size - Global - Size. The Global Element Sizes window opens. Input 1.5 in the SIZE box and click OK button.
6). Meshing:
From Main Menu click Preprocessor - Meshing - Mesh - Areas - Free. The Mesh Areas window opens. Pick the area and click OK.
The warning window appears due to the existence of six singular elements. Click Close button and proceed to the next operation below.
Figure below shows the plate area meshed by ordinary 8-node isoparametric finite elements except for the vicinity of the crack tip where we have six singular elements.
Figure below is an enlarged view of the singular elements around the crack tip showing that six triangular elements are placed in a radial manner and that the size of the second row of elements is half the radius of the first row of elements, i,e., triangular singular elements.
7). Boundary Conditions:
Due to the symmetry, the constraint conditions of the quarter plate model are UX-fixed condition on the left end side, that is, the line between key points 4 and 5, and UY-fixed condition on the bottom side of the quarter plate model.
From Main Menu click Solution - Define Loads - Apply - Structural - Displacement - On Lines. The Apply U, ROT on Lines window opens. Confirming that the Pick and Single buttons are selected, move the upward arrow onto the line between key points 4 and 5 and click the left button of the mouse.
Click the OK button to display another Apply U, ROT on Lines window. Select UX and click OK.
Again from Main Menu click Solution - Define Loads - Apply - Structural - Displacement - On Lines. The Apply U, ROT on Lines window opens. Confirming that the Pick and Single buttons are selected, move the upward arrow onto the bottom line and click the left button of the mouse.
Click the OK button to display another Apply U, ROT on Lines window. Select UY and click OK.
8). To apply uniform longitudinal stress on the right end of the quarter plate model,
From Main Menu click Solution - Define Loads - Apply - Structural - Pressure - On Lines. Apply PRES on Lines window opens and the upward arrow appears when the mouse cursor is moved to the ANSYS Graphics window. Pick the right-end side of the quarter plate area and click OK.
Another Apply PRES on Lines window opens. Select Constant value in the Apply PRES on Lines as a box and input -10 (MPa) in the VALUE Load PRES value box and leave a blank in the value box. Click OK to define a uniform tensile stress of 10 MPa applied to the right end of the quarter plate model.
9). Solution Procedures:
From Main Menu click Solution - Solve - Current LS. The Solve Current Load Step and /STATUS Command windows appear.
Click OK button in the Solve Current Load Step window to begin the solution of the current load step. The Verify window opens . Proceed to the next operation below by clicking Yes button in the window.
When the solution is completed, the Note window appears. Click the Close button to close the Note window.
10). Contour Plot of Stress:
From Main Menu click General Postproc - Plot Results - Contour Plot - Nodal Solution. The Contour Nodal Solution Data window opens. Select Stress and X-Component of stress. Then select Deformed shape with deformed edge in the Undisplaced shape key box to compare the shapes of the cracked plate before and after deformation. Then click OK.
X-Component of stress:
Figure below is an enlarged view of the longitudinal stress distribution around the crack tip showing that very high tensile stress is induced at the crack tip whereas almost zero stress around the crack surface and that the crack shape is parabolic.
11). Zoom the crack tip region. From Menu click PlotCtrls - Pan Zoom Rotate...
Pan-Zoom-Rotate window opens.
In the window click on the Win Zoom button and zoom the crack-tip region. Then click on the Close button to close the window.
To plot the node only, from Menu click Plot - Nodes.
Turn on the node numbering by selecting Menu - PlotCtrls - Numbering... then click the box for 'Node numbers' then finally click on OK.
For each node its relevant number appears.
To define Crack-Face Path, from Main Menu click General Postproc - Path Operations - Define Path - By Nodes. Pick the crack tip node (node #606), then the quarter-point node (node #669), and finally the third node (node #668) on the crack face and click OK.
The By Nodes window appears. Enter K1 for Define Path Name then click OK.
Close the Path Command window.
To define Local Crack-Tip Coordinate System, from Menu click WorkPlane - Local Coordinate Systems - Create Local CS - By 3 Nodes.
Create CS by 3 Nodes window opens. Pick node #606 (the crack-tip node), then node #670 and finally node #5516.
Create CS by 3 Nodes window opens. From the window that the reference number of the crack-tip coordinate system is 11, click on OK button.
To activate the Local Crack-Tip Coordinate System, from Menu click WorkPlane - Change Active CS to - Specified Coord Sys ...
The Change Active CS to Specified CS window opens. Enter 11 for Coordinate System number and click OK.
To activate the crack-tip coordinate system, from Main Menu click General Postproc - Options for Outp. Options for Output window opens. In the window select Local system for Results coord system and enter 11 for Local system reference no and click OK.
To determine the Mode-I Stress Intensity Factor using KCALC, from Main Menu click General Postproc - Nodal Calcs - Stress Int Factr. Stress Intensity Factor window opens. In the below window select 'Plain Strain' for 'Disp extrapolate based on' and 'Half-symm b.c.' for 'Model Type'.
Pan-Zoom-Rotate window opens.
In the window click on the Win Zoom button and zoom the crack-tip region. Then click on the Close button to close the window.
To plot the node only, from Menu click Plot - Nodes.
Turn on the node numbering by selecting Menu - PlotCtrls - Numbering... then click the box for 'Node numbers' then finally click on OK.
For each node its relevant number appears.
To define Crack-Face Path, from Main Menu click General Postproc - Path Operations - Define Path - By Nodes. Pick the crack tip node (node #606), then the quarter-point node (node #669), and finally the third node (node #668) on the crack face and click OK.
The By Nodes window appears. Enter K1 for Define Path Name then click OK.
Close the Path Command window.
To define Local Crack-Tip Coordinate System, from Menu click WorkPlane - Local Coordinate Systems - Create Local CS - By 3 Nodes.
Create CS by 3 Nodes window opens. Pick node #606 (the crack-tip node), then node #670 and finally node #5516.
Create CS by 3 Nodes window opens. From the window that the reference number of the crack-tip coordinate system is 11, click on OK button.
To activate the Local Crack-Tip Coordinate System, from Menu click WorkPlane - Change Active CS to - Specified Coord Sys ...
The Change Active CS to Specified CS window opens. Enter 11 for Coordinate System number and click OK.
To activate the crack-tip coordinate system, from Main Menu click General Postproc - Options for Outp. Options for Output window opens. In the window select Local system for Results coord system and enter 11 for Local system reference no and click OK.
To determine the Mode-I Stress Intensity Factor using KCALC, from Main Menu click General Postproc - Nodal Calcs - Stress Int Factr. Stress Intensity Factor window opens. In the below window select 'Plain Strain' for 'Disp extrapolate based on' and 'Half-symm b.c.' for 'Model Type'.
Click on OK. Another window which shows the SIFs at the crack tip.
KI = 57.682, KII = 0, and KIII = 0.
Note that for this problem, tabulated solutions for the mode-I SIF KI are available in the literature. For example, an analytical solution given by W.D. Pilkey (Formulas for Stress, Strain, and Structural Matrices) is:
KI = 57.682, KII = 0, and KIII = 0.
Note that for this problem, tabulated solutions for the mode-I SIF KI are available in the literature. For example, an analytical solution given by W.D. Pilkey (Formulas for Stress, Strain, and Structural Matrices) is:
For a = 10mm, b = 100mm and σ = 10 MPa and use of this solution yields KI = 56.03.
The ANSYS solution for KI (57.682) is in very good agreement with that obtained from W.D. Pilkey (56.03). The discrepancy is:
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