Signature: UbuJ. Z l / ^ ^ ^ c 2 ^ _. Date: ... A 3-D digital image correlation system capable of full .... Figure 5.1 Flow Chart for HTS Inverse Solution Method. 70.
ELASTIC PROPERTIES OF SANDWICH COMPOSITE PANELS USING 3-D DIGITAL IMAGE CORRELATION WITH THE HYDROMAT TEST SYSTEM By Paul Thomas Melrose B.S. University of Maine, 2002
A THESIS Submitted in Partial Fulfillment of the Requirements for the Degree of Master of Science (in Mechanical Engineering)
The Graduate School The University of Maine August, 2004
Advisory Committee: Roberto Lopez-Anido, Ph.D., P.E., Associate Professor, Department of Civil and Environmental Engineering Senthil Vel, Ph.D. Assistant Professor, Department of Mechanical Engineering Michael Peterson, Ph.D. Associate Professor, Department of Mechanical Engineering
© 2004 Paul Thomas Melrose All Rights Reserved
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LIBRARY RIGHTS STATEMENT
In presenting this thesis in partial fulfillment of the requirements for an advanced degree at The University of Maine, I hereby agree that the Library shall make it freely available for inspection. I further agree that permission for "fair use" copying of this thesis for scholarly purposes may be granted by the Librarian. It is understood that any copying or publication of this thesis for financial gain shall not be allowed without my written permission.
Signature: Date:
UbuJ
Z l / ^ ^ ^
c 2
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ELASTIC PROPERTIES OF SANDWICH COMPOSITE PANELS USING 3-D DIGITAL IMAGE CORRELATION WITH THE HYDROMAT TEST SYSTEM
By: Paul Thomas Melrose Thesis Advisor: Dr. Roberto Lopez-Anido
An Abstract of the Thesis Presented in Partial Fulfillment of the Requirements for the Degree of Master of Science (in Mechanical Engineering) August, 2004 The hydromat test system (HTS), ASTM D6416, is a recent experimental approach to characterize the structural performance of sandwich composite panels. In this thesis, hydromat test panels were modeled using sandwich composite plate theory with a Navier's solution form. The complete methodology used to predict the response of a hydromat test panel is presented. In previous work dealing with hydromat tests of sandwich structures, the sandwich plate theory was oversimplified and therefore a formal application of classical lamination theory was not available. The analysis process from classical lamination theory to sandwich plate theory is illustrated and the solution is compared to the results from a finite element method (FEM) code. It was found that the proposed model overcomes the limitations of the existing HTS solutions and correlates very well with FEM results. A combined experimental and numerical approach for determining bending and shear stiffness parameters of a sandwich composite panel evaluated with the hydromat test was developed. A method for computing the bending and shear stiffness parameters
of an orthotropic sandwich composite panel using the hydromat test system is presented and verified. The method is considered to be robust for two reasons. First, it was found that when random additive noise was introduced into simulated field data, convergence to the known solution was attained. Second, it was found that the solution converged for initial seed values containing significant error. To implement the inverse solution method experimentally, full-field strain and displacement data was required. A 3-D digital image correlation system capable of full field measurements of strains and displacements was utilized with the hydromat test system. The accuracy and precision of the 3-D digital image correlation system was quantified to allow use with the hydromat test system ensuring reliable measurements. The stiffness parameters of the sandwich composite plate were optimized based on the experimental displacement and in-plane strain measurements from image correlation. This combined experimental and numerical solution method when applied to E-glass balsa wood sandwich panels produced reliable results. The relevance of this combined experimental-numerical method is that it could change the way composite material structures are evaluated, since it may be possible to determine relevant stiffness parameters from plate bending tests. This approach reduces the experimental effort required to characterize sandwich composite panels, while improving the reliability of the predictions.
ACKNOWLEDGMENTS The research presented in this Thesis was sponsored by the National Science Foundation (NSF) through the CAREER Grant No. CMS-0093678.
My Advisory Committee Dr. Roberto Lopez-Anido, Associate Professor, Department of Civil and Environmental Engineering, University of Maine Dr. Senthil Vel, Assistant Professor, Department of Mechanical Engineering, University of Maine Dr. Michael Peterson, Associate Professor, Department of Mechanical Engineering, University of Maine
Other Thanks Kasey Wyman, My Fiancee Dr. Lech Muszynski, Assistant Scientist, Advanced Engineered Wood Composites Center, University of Maine.
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TABLE OF CONTENTS ACKNOWLEDGMENTS
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LIST OF TABLES
ix
LIST OF FIGURES
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1. INTRODUCTION
1
1.1. Motivation and Objectives
1
1.2. Fabrication of Sandwich Composite Panels by VARTM/SCRIMP
4
1.3. Implementation of a Plate Bending Test Setup Under Uniform Load the Hydromat Test System
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1.4. Validation of an Orthotropic Sandwich Composite Panel Model
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1.5. Inverse Method to Obtain Panel Stiffness Parameters Based On Numerical Optimization Techniques
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1.6. Accuracy and Precision of 3-D Digital Image Correlation System for Material Tensile Tests
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1.7. Combined Numerical and Experimental Approach to Obtain Sandwich Composite Panel Stiffness Properties 1.8. References Cited
7 8
2. FABRICATION OF SANDWICH COMPOSITE PANELS BY VARTM/SCRIMP
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2.1. Abstract
9
2.2. Introduction
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2.3. Sandwich Panel Specifications
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2.4. Sandwich Composite Panel Fabrication Method
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2.4.1. Initial Setup
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2.4.2. Infusion Process
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2.4.3. Infusion
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2.5. Conclusions and Recommendations
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2.6 References Cited
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3. IMPLEMENTATION OF A PLATE BENDING TEST SETUP UNDER UNIFORM LOAD HYDROMAT TEST SYSTEM
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3.1. Abstract
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3.2. Introduction
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3.3. Test Setup
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3.4. Hydromat Modifications
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3.4.1. Small Bladder
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3.4.2. Dynamic Loading Inertial Compensation
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3.4.3. 3-D Digital Image Correlation
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3.5. Sandwich Panel Response Verification
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3.5.1. Carbon Composite Properties
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3.5.2. Core Shear Properties
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3.5.3. Sandwich Panel Fabrication
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3.5.4. Modified Hydromat Testing
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3.5.5. Modeling Predictions versus Experimental Results
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3.5.6. Experimental Failure Modes
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3.6. Conclusions and Recommendations
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3.7. References Cited
41
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4. VALIDATION OF AN ORTHOTROPIC SANDWICH COMPOSITE PANEL MODEL
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4.1. Abstract
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4.2. Hydromat Testing Introduction
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4.3. Sandwich Plate Overview
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4.4. Analytical Solution
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4.4.1. Classical Lamination Theory
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4.4.2. Sandwich Plate Solution as Applied to the Hydromat Test System
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4.4.3. Previous Hydromat Orthotropic Sandwich Composite Plate Solution
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4.4.4. Finite Element Modeling Compared to Analytical Results
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4.5. Conclusions
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4.6. Symbols for Hydromat Modeling
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4.7 References Cited
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5. INVERSE METHOD TO OBTAIN PANEL STIFFNESS PARAMETERS BASED ON NUMERICAL OPTIMIZATION TECHNIQUES
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5.1. Abstract
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5.2. Introduction
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5.3. Inverse Solution Method
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5.4. Verification
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5.5. Conclusions and Recommendations
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5.6 References Cited
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6. ACCURACY AND PRECISION OF 3-D DIGITAL IMAGE CORRELATION SYSTEM FOR MATERIAL TENSILE TESTS
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6.1. Abstract
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6.2. Literature Review
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6.3. Introduction To 3-D Digital Image Correlation
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6.4. Significance
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6.5. Summary of Test Method
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6.5.1 Extensometer Based Coupon Testing
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6.5.2 Resistive Strain Gage Based Coupons
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6.6. Material Property Calculations
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6.6.1. Extensometer
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6.6.2. Digital Image Correlation
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6.6.3. Resistive Strain Gage
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6.7. Results
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6.7.1. Extensometer Based Coupon Testing Set
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6.7.2. Resistive Strain Gage Based Coupon Testing
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6.7.3. Digital Image Correlation Based Coupon Testing
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6.8. Discussion of Results
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6.9. Conclusions and Recommendations
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6.10. References Cited
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7. COMBINED NUMERICAL AND EXPERIMENTAL APPROACH TO OBTAIN SANDWICH COMPOSITE PANEL STIFFNESS PROPERTIES
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7.1. Abstract
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7.2. Introduction
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7.3. Inverse Solution Method
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7.4. Hydromat Testing
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7.5. Combined Experimental and Numerical Approach
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7.6. Conclusions and Recommendations
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7.7. References Cited
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8. CONCLUSIONS, PRACTICAL RECOMMENDATIONS AND FUTURE RESEARCH WORK
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8.1. Fabrication of Sandwich Composite Panels By VARTM/SCRIMP
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8.2. Implementation of a Plate Bending Test Setup under Uniform Load Hydromat Test System
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8.3. Validation of an Orthotropic Sandwich Composite Panel Model
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8.4. Inverse Method to Obtain Panel Stiffness Parameters Based on Numerical Optimization Techniques
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8.5. Accuracy and Precision of 3-D Digital Image Correlation System for Material Tensile Tests
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8.6. Combined Numerical and Experimental Approach to Obtain Sandwich Composite Panel Stiffness Properties
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8.7. Practical Recommendations
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8.8. Future Research Work
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8.9. References Cited
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APPENDIX. Matlab Code
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BIOGRAPHY OF THE AUTHOR
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LIST OF TABLES Table 2.1 Fabrication Comparison
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Table 2.2 Typical Properties of Derakane 8084 (Dow Chemical)
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Table 2.3 Sandwich Panel Layup
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Table 2.4 Typical Gel Times for Resin Mixtures (Dow Chemical)
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Table 3.1 Mechanical Properties of Carbon Fiber Composite Sheets
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Table 3.2 Manufacturer's Properties for Lightweight Core Materials
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Table 3.3 Computed Core Material Properties (ASTM D638)
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Table 3.4 Sandwich Plate Modeling Predictions versus Experimental Values
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Table 4.1 FEM model Description
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Table 4.2 Physical Constants and Elastic Properties for Hydromat Test Modeling
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Table 4.3 Analytical Sandwich Plate Solution versus Finite Element Modeling
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Table 4.4 MTU Sandwich Plate Solution versus Finite Element Modeling
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Table 5.1 Optimization Routine Input Data from Full Field Data
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Table 5.2 Physical Constants and Elastic Properties for Hydromat Test Model
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Table 5.3 Optimization Results (No Noise)
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Table 5.4 Optimization Results (25% Noise)
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Table 5.5 Optimization Results (50% Noise)
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Table 6.1 Test Results for Extensometer Derived Properties versus IC Derived Properties
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Table 6.2 Bi-Axial Strain Gage Data with Transverse Sensitivity Correction versus IC
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Table 6.3 Axial Strain Gage Derived Properties versus IC Derived Properties
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Table 6.4 Bi-Axial Strain Gage Derived Properties versus IC Derived Properties Data
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Table 7.1 Bending and Shear Stiffness Coefficients: Seed Value
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Table 7.2 Bending and Shear Stiffness Coefficients for Sandwich Panel with Balsa Wood Core: Computed Values
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Table 7.3 Bending and Shear Stiffness Coefficients for Sandwich Panel with PVC Foam Core: Computed Values
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LIST OF FIGURES Figure 2.1 VARTM/SCRIMP Preform Profile
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Figure 2.2 Panel Fabrication Layout
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Figure 2.3 Panel Fabrication Layout
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Figure 2.4 Resin Flow Front
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Figure 3.1 Hydromat Test System
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Figure 3.2 500 mm Hydromat Pressure Bladder
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Figure 3.3 Hydromat Test Frame (lower support frame omitted)
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Figure 3.4 Line Load Diffuser Strips (Plan View)
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Figure 3.5 Sandwich Plate Section and Edge Support Detail
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Figure 3.6 Carbon Balsa Load versus Strain X
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Figure 3.7 Carbon Foam Load versus Strain X
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Figure 3.8 Carbon Balsa Load versus Transverse Displacement
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Figure 3.9 Carbon Foam Load versus Transverse Displacement
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Figure 3.10 Typical Balsa Shear Failure Mode in ASTM C273
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Figure 3.11 Core Shear Failure Mode of the Balsa Carbon Panel
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Figure 3.12 Typical Core Transverse Shear Stress Distributions xxz and xyz
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Figure 3.13 Predicted Core Shear Stress Distribution for the Carbon Balsa Panel
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Figure 4.1 Hydromat Test Frame (lower support frame omitted)
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Figure 4.2 Support Span and Pressure Patch
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Figure 4.3 Sandwich Plate Section and Edge Support Detail
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Figure 4.4 Lamina Coordinate System
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Figure 4.5 Sandwich Panel Section with z-Coordinates
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Figure 4.6 FEM Convergence Study
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Figure 5.1 Flow Chart for HTS Inverse Solution Method
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Figure 5.2 Simulated Transverse Displacement Data with 50% Noise
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Figure 5.3 Optimization Solution Error versus Random Noise Level
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Figure 6.1 Increase in Facet Size
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Figure 6.2 Increase in Facet Step
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Figure 6.3 Increase in the Calculation Base (Matrix Size)
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Figure 6.4 Increase in the Precision versus Facet Size, with a Facet Step of 13 and a 3x3 Calculation Base
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Figure 6.5 Increase in the Precision versus Facet Step, with a Facet Size of 15 and a 3x3 Calculation Base
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Figure 6.6: Increase in the Precision versus Calculation Base (Matrix Size) with a Facet Size of 15, and a Facet Step of 13 Figure 6.7 Dimensions of the aluminum tensile coupons
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Figure 6.8 Painted Tensile Coupon with Resistive Foil Strain Gage (gage is painted)
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Figure 6.9 Tensile Test Setup with Digital Image Correlation System
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Figure 6.10 Bi-Axial and Axial Strain Gage Placement
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Figure 6.11 Thickness Contoured Plot obtained with IC Showing the Resistive Foil Strain Gage (Scale: 0 mm - 0.085mm)
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Figure 6.12 Axial Stress versus Axial Strain Curve (Extensometer Based)
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Figure 6.13 Linearly Fitted Residuals from Figure 1 (0-150MPa Stress Range)
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Figure 6.14 Linearly Fitted Residuals from Figure 1 (0-400 MPa Stress Range)
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Figure 6.15 Typical Axial Stress versus Axial Strain Curve (IC Based)
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Figure 6.16 Typical Axial Stress versus Transverse Strain Curve (IC Based)
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Figure 6.17 Typical Stress versus Strain Curve Using a Bi-Axial Resistive Foil Strain Gage
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Figure 6.18 Typical Stress versus Strain Curve Using a Bi-Axial Resistive Foil Strain Gage
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Figure 7.1 Hydromat Test System
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Figure 7.2 500 mm Hydromat Pressure Bladder
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Figure 7.3 Contour plots of hydromat panel (transverse displacements)
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Figure 7.4 Contour plots of hydromat panel (in-plane shear strains)
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1. INTRODUCTION 1.1 Motivation and Objectives Sandwich composite panels are made from high stiffness face sheets and low density core materials. This efficient structural element can be modeled using classical lamination theory combined with first-order shear deformation (FSDT) theory. The modeling approach requires that the components of the stiffness matrix be defined, thus determining the global response of the sandwich composite structure. The required stiffness parameters can be computed based on the elastic properties of the constituents, face-sheets and core, along with thicknesses and the material directions. The elastic properties can be obtained from material coupon tests of the individual constituents. However, there are a number of considerations in sandwich composite panel fabrication that may limit the validity of the modeling approach based on coupon tests. For example, the stiffness parameters can be affected by actual thickness of face-sheets in sandwich panels, which may differ from measured thickness in material test coupons. In addition, the resin used in the face-sheets, both as binder for fiber reinforcement and an adhesive to the core, may permeate into the core material resulting in non homogeneous properties. The core material properties may also vary spatially, which is typical of natural materials such as end-grain balsa wood. Other fabrication issues such as fiber alignment in multidirectional face-sheets can also affect stiffness parameters. It should be noted that conducting material coupon tests of relatively heavy woven-roving fabric reinforcement, typically used for marine and infrastructure sandwich construction applications, presents a number of challenges. Current ASTM standard test procedures were designed for relatively thin laminates made of unidirectional composite
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laminas (e.g., prepregs). For example, typical coupon widths and gage lengths in current standards need to be modified for composites reinforced with woven-roving. In a similar way, obtaining shear properties of core materials present a series of practical challenges. For example, when testing cellular core materials (e.g., end-grain balsa wood or synthetic honeycomb) the shear strength is affected by specimen thickness. Therefore, the validity of applying coupon properties for predicting sandwich composite panel properties may be compromised due to material variability, fabrication effects and limitations of current material coupon test procedures. To address these issues, a new method to obtain stiffness parameters and characterize the structural response of sandwich composite panels was needed. To accomplish this goal, a test method with the following features was required: large specimen geometry, uniformly distributed applied load, well defined boundary conditions and repeatable procedure. The hydromat test system (HTS), which is specified by ASTM D6416, addressed all of the relevant issues and, therefore, it was selected as the preferred experimental method for characterizing sandwich composites in this thesis. Two limitations were found when conducting the HTS with sandwich composite panels. First, the existing orthotropic sandwich composite plate closed-form solution was oversimplified. The oversimplification resulted in erroneous deflection and strain predictions when comparing with a finite element analysis. Second, the method requires using bonded resistive foil gages and a linear variable displacement transducer (LVDT), which provided limited information concerning a panel's response. To correct the predictive model and to enhance the HTS method a formal Navier closed-form solution based on classical lamination theory and FSDT was implemented
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and validated with finite element analysis. The classical textbook by Whitney (1987) was used as a guideline for the derivations. The derived closed-form solution is valid for specially orthotropic and symmetric sandwich composite plates. To improve the experimental measurements during hydromat testing an innovative non contact full field measuring technique, 3-D digital image correlation, was chosen to acquire strains and displacements. The accuracy and precision of the 3-D digital image correlation system was evaluated to ensure that the system would perform comparably to the foil strain gages and the LVDT being replaced. 3-D digital image correlation improved the potential of HTS to characterize the structural response of sandwich composite panels. Furthermore, the acquired data fields help spur the development of an inverse solution method to obtain stiffness parameters from hydromat tests. A combined experimental and numerical approach for determining bending and shear stiffness parameters of a sandwich composite panel was developed. The approach correlated experimental hydromat tests to the developed predictive modeling for the HTS. As a starting point, the predictive model with micromechanics equations for elastic properties was used to estimate the response of the sandwich composite panel during the hydromat test. The panel was then experimentally evaluated using the hydromat test system equipped with the 3-D digital image correlation system. Differences between the predicted response and the actual panel response became apparent. Using numerical optimization techniques, the stiffness parameters defining the predictive model were adjusted until the model solution was the best possible fit to the experimental data fields generated using the digital image correlation. This method improved the quality of the
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stiffness parameters and generated more reliable results than the previous methods used to characterize sandwich composites. The relevance of this combined experimental-numerical method is that it could change the way composite material structures are evaluated, since it may be possible to determine relevant stiffness parameters from plate bending tests. This approach reduces the experimental effort required to characterize sandwich composite panels, while improving the reliability of the predictions. 1.2. Fabrication of Sandwich Composite Panels by VARTM/SCRIMP The primary aspects of the fabrication process selected for sandwich composite panels are presented in Chapter 2. The sandwich panels are fabricated using the Vacuum-Assisted Resin Transfer Molding (VARTM) process with the proprietary Seemann Composites Resin Infusion Molding Process (SCRIMP) technology. In the VARTM/SCRIMP method the face sheets of a sandwich composite panel are infused and bonded to the core material in one step. This leads to significant advantages over traditional fabrication methods where the face sheet is bonded to the core in a secondary operation. The infusion method is presented by describing the fabrication of four sandwich composite panels. The materials selected for the composite face-sheets, which are typical of marine and construction applications, were woven-roving E-glass and vinyl ester resin. Two core materials representative of sandwich construction applications were selected: polyvinyl chloride (PVC) foam and end-grain balsa wood. The presented resin infusion method is general and can be used as a guideline to fabricate more complex sandwich composite structures.
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1.3. Implementation of a Plate Bending Test Setup Under Uniform Load the Hydromat Test System The use of sandwich composite panels is becoming increasingly widespread in transportation and marine structures due to their high strength to weight ratio and corrosion resistance. With the growing market for sandwich panels, new comprehensive experimental methods are needed to characterize the structural performance. The experimental method should account for the actual properties of sandwich panel constituents, face-sheets and core, as well as the effects of the fabrication process. The hydromat test system, ASTM D6416 (ASTM 2001), is a recent experimental approach used to characterize sandwich composite panels. Chapter 3 introduces the hydromat test system (HTS) and illustrates how the HTS can be used to verify the structural performance of sandwich composite panels. 1.4. Validation of an Orthotropic Sandwich Composite Panel Model Chapter 4 provides an overview of classical lamination theory, FSDT and sandwich plate theory as it applies to hydromat test panels. The chapter presents the complete methodology used to predict the response of a hydromat test panel and foreshadows future inverse solution methods. As discussed earlier, in previous works dealing with sandwich structures and hydromat testing, the sandwich plate theory has been oversimplified and therefore a formal application of classical lamination theory and FSDT has not been implemented. Chapter 4 illustrates the solution process from classical lamination theory and FSDT to sandwich plate theory and compares the solution to finite element modeling (FEM) results.
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1.5. Inverse Method to Obtain Panel Stiffness Parameters Based on Numerical Optimization Techniques A combined experimental and numerical approach for determining bending and shear stiffness parameters of a sandwich composite panel evaluated with the hydromat test was developed. Currently, a solution is available for determining the bending and shear stiffness parameters of an isotropic sandwich panel when tested with the hydromat test system (ASTM 2001). Chapter 5 presents and verifies a method for computing the bending and shear stiffness parameters of an orthotropic sandwich composites panel using the hydromat test system. The solution was verified using simulated full-field data. A set of initial seed values, which were 40 % lower than the known stiffness parameters, were considered to verify convergence. Also, the known displacement and strain field solution were perturbed using additive random noise. The robustness of the optimization method was demonstrated through the convergence of the perturbed solution. 1.6. Accuracy and Precision of 3-D Digital Image Correlation System for Material Tensile Tests A digital image correlation (IC) system capable of measuring 3-D displacements and in plane strain fields was evaluated. Initially, a limitation of the IC system was that the accuracy and precision of the system when compared to traditional strain measurement techniques was unknown. In Chapter 6, the accuracy and precision of the 3-D digital image correlation system were correlated to measurements from resistive strain gages and an extensometer. To quantify the accuracy and precision of the IC system a set of tensile tests were performed. The material selected for the coupons was 7075 T651, a stress relieved
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aluminum alloy. The aluminum alloy 7075 T651 was selected because of its consistent material properties. Three sets of coupons, each consisting of eight tensile specimens, were considered. The first set of coupons was instrumented with an extensometer, the second and third coupon sets were instrumented with resistive strain gages. During tensile tests, each coupon was monitored using the digital image correlation system allowing comparison to the conventional instrumentation. Material properties for the test coupons were determined using data from the IC system, resistive strain gages and the extensometer. Variations in the material properties defined the precision of the systems. The accuracy of the IC system was determined by comparing it to the extensometer, which was calibrated and traceable to the National Institute of Standards and Technology (NIST). Additionally, the accuracy of all systems was determined by the comparing the computed material properties to published values for the aluminum alloy selected. 1.7. Combined Numerical and Experimental Approach to Obtain Sandwich Composite Panel Stiffness Properties A method to determine elastic properties of sandwich composite panels through experimental and analytical techniques is presented in Chapter 7. The 3-D digital image correlation system that allows for full field measurements of strains and displacements was utilized in conjunction with the hydromat test system. Experimental full-field displacement and strain measurements from the hydromat test were correlated to the composite sandwich plate solution through numerical optimization techniques. Based on this correlation, an inverse analysis technique was implemented to compute elastic properties of the sandwich panel components (face sheets and core). Numerical
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techniques were utilized to minimize the error in the prediction of elastic properties. This test method shows promising results based on the stiffness characterization from two sets of test panels evaluated. This numerical-experimental method can be used to determine all the relevant stiffness parameters of a sandwich composite panel from a plate bending test. From the four bending stiffness parameters (Dn, D12, D22 and D66) and based on sandwich plate theory assumptions, it is possible to compute the four elastic properties that characterize an individual lamina (Ei, E2, G12, V12) of the composite face sheet. Also from the two transverse shear stiffness parameters (A55, A44) and based on sandwich plate theory assumptions, it is possible to compute the two shear moduli that characterize an orthotropic core (Go, G23). Another application of the method is to assess the effect of constituent material variability in a relatively large test specimen (i.e., an HTS panel area is typically more than 40 times larger than a material test coupon). In the same way, the effect of the fabrication process, such as thickness consolidation of composite face-sheets and resin permeation into the core can be assessed. 1.8. References Cited ASTM (2001). D6416 Standard Test Method for Two-Dimensional Flexural Properties of Simply Supported Sandwich Composite Plates Subjected to a Distributed Load. West Conshohocken, PA, American Society of Testing and Materials. Whitney, J. M. (1987). Structural Analysis of Laminated Anisotropic Plates. Lancaster, PA, Technomic Publishing Company, Inc.
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2. FABRICATION OF SANDWICH COMPOSITE PANELS BY VARTM/SCRIMP 2.1. Abstract The primary aspects of the fabrication process of E-glass / vinyl ester sandwich composite panels are presented. The sandwich panels are fabricated using the Vacuum Assisted Resin Transfer Molding (VARTM) process with the proprietary SCRIMP technology. In the VARTM/SCRIMP method the face sheets of a sandwich composite panel are infused and bonded to the core material in one step. This leads to significant advantages over traditional fabrication methods where the face sheet is bonded to the core in a secondary operation. The infusion method is presented by describing the fabrication of four sandwich composite panels. The presented infusion method is general and can be used as a guideline to fabricate more complex sandwich composite structures. 2.2. Introduction Sandwich composite panels are constructed by bonding thin high stiffness face sheets to a low stiffness but relatively thick core material. As the core material becomes thicker the sandwich panel becomes stiffer. Typical sandwich constructions exhibit a high stiffness under transverse loads when compared to materials with similar weight. The high stiffness to weight ratio makes sandwich construction a very attractive design option in weight critical structures. The goal of this chapter is to develop a process to fabricate high quality sandwich composite panels that can be evaluated using the hydromat test system (Bertelsen 1992; Bertelsen 1994; Bertelsen 2000). Traditional composite fabrication methods range from the wet layup techniques, used in the marine industry for many years, to the pre preg
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technologies typically found in the aerospace industries. The VARTM/SCRIMP method presented in this chapter lies somewhere between the hand layup process and prepreg technology. The advantages and disadvantages of each fabrication process are shown in Table 2.1. Table 2.1 Fabrication Comparison Fabrication Process
Advantages
Disadvantages
VARTM/SCRIMP
High fiber volume fraction Low void content, Cost effective
Involved process Proprietary process
Hand Lay-up
Simple process
Low fiber volume fraction High void content VOC emissions
Pre Preg
High fiber volume fraction, Low void content Consistent material properties
Requires an autoclave Expensive
The VARTM/SCRIMP resin infusion process has been used to fabricate a variety of parts ranging from solid composite laminates to sandwich composite structures. By fabricating sandwich composite panels with the SCRIMP technologies, the typical twostep process of bonding the face sheets to the core material can be avoided (Smith 2001). Thus, a more consistent part with less labor is produced. 2.3. Sandwich Panel Specifications A sandwich composite panel is constructed from two distinct constituents, face sheets and core. The sandwich composite face sheets were constructed from a fiberreinforced polymer (FRP) composite with a core material of either end-grain balsa wood or cross-linked polyvinyl chloride (PVC) foam. 10
The sandwich panel layups were designed to create specific panel responses for the plate bending tests performed in Chapter 7. The balsa cored panels should exhibit a dominant bending response and should fail on the compressive face sheet when tested in accordance with ASTM D6416. The foame core panels should exhibit an approximately equal response in shear and bending and should fail in core shear failure when tested with ASTMD6416. The sandwich composite panel face sheets were constructed from E-glass reinforcement set in a vinyl ester polymer. This face sheet configuration created a transparent face sheet that can be visually inspected for defects where as carbon fiber face sheets are opaque. The fiber reinforcement was woven roving with a weight per unit area of 814 kg/m2 (24 oz/yd2) and the style was Vetrotex 324. The warp and fill directions have 55% and 45%, of the total fiber weight respectively. The face sheets consisted of either one or two layers of fabric reinforcement where the warp direction corresponds to the principal material direction 1. The polymer resin was Derakane 8084, which is an elastomer-modified vinyl ester with properties listed in Table 2.2. Table 2.2 Typical Properties of Derakane 8084 (Dow Chemical) Typical Properties'
Dynamic Viscosity @ 25°C (77°F), mPa.s Styrene Content, % Density @ 25°C (77°F), g/ml
360 40 1.02
The core materials selected for the sandwich composite panels were end-grain balsa wood and cross-linked PVC foam. Panel set A, the sandwich panels constructed with an end-grain balsa wood core, had one layer face sheets while panel set B were designed with two layer face sheets to increase the bending rigidity. The end-grain balsa
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core was Baltek D100 rigid sheet supplied with the proprietary AL600/10 coating for decreasing porosity and increasing bond strength. The PVC foam was supplied by Diab and is designated as Divinycell H80. Both core materials were rigid sheets with a 25 mm thickness. The density of the core end-grain balsa was 150 kg/m3 while the PVC foam had a density of 80kg/m3. The sandwich panel layup for each panel type is presented in Table 2.3. The two core materials, designated as D100 and H80, correspond to the endgrain balsa wood and the cross-linked PVC foam respectively. Table 2.3 Sandwich Panel Layup Sandwich Panel Type Panel Type A Panel Type B
Layup [0, (D100), 0] [0, 90, (H80), 90, 0]
The materials selected for the sandwich panels described above are representative of typical material used in the marine industry. The elastomer modified vinyl ester resin is currently used by the United States Navy and was implemented in the construction of the Swedish Navy vessel the Visby. The Vetrotex E-glass woven roving is currently being studied by the United States Navy. The PVC foam core was used in the construction of the Visby Corvette Class and the end grain balsa core has been used in various marine structures although the United States Navy only recommends it for topside use. It is important to note that end grain balsa has been successfully used below the waterline for commercial applications. This can be done as long as the end grain balsa is scored and infused with SCRIMP. This configuration creates a cell of balsa that in encased with resin, thus preventing the ingress of moisture that typically destroys balsa cored sandwich constructions.
12
2.4. Sandwich Composite Panel Fabrication Method 2.4.1. Initial Setup The VARTM/SCRIMP fabrication process consists of layering several materials, as illustrated in Figure 2.1.
Figure 2.1 VARTM/SCRIMP Preform Profile The layers were carefully placed on a flat molding surface. A stainless steel table was used for initial trial fabrications. However, it was later found that a sheet of glass was more beneficial, as it allowed for an improved surface finish and provided a larger fabrication area than the stainless steel table. Before placing the layers, the mold surface was cleaned using a degreaser. This included spraying the degreaser on the surface and wiping it down with a clean cloth, being careful not to wipe the same spot twice, in order to prevent recontamination. Once the surface had dried, vacuum sealant tape (tacky tape) was applied to the surface. The tacky tape was used to adhere the vacuum bag to the mold surface, which was one of the final steps of assembly. However, the tacky tape was placed at this stage in order to give an outline of the desired fabrication area. The
13
fabrication area needs to encompass the part and typically allows for an additional clearance of 100 mm to 150 mm on all four sides. As a precautionary measure, a release agent was applied to the working surface preventing resin from bonding to the mold surface during fabrication. As a secondary measure, a release film was placed on the surface to aid in the removal of the part from the mold surface. This step was most critical when using the stainless steel surface. But, when using the glass surface this layer can be omitted since removing cured resin from the glass is quite easy with the release agent alone. The lower flow media, a 60% greenhouse shade cloth, was placed on top of the treated surface and release film, as shown in Figure 2.1. The flow media serves as a flow path for the resin. Therefore, it was important to align the media such that the strands of the mesh are traveling in the same direction as the desired flow path. It was also important to consider the length of the flow path between resin feed lines. The typical maximum length was 610 mm between resin feeder lines but may vary depending on resin viscosity and fiber reinforcement thickness. Typically, if the flow media extends over more than 610 mm without an addition of a feeder line or if the part has many layers of reinforcement the speed of the resin flow will be reduced, forcing the resin to catalyze before the part is fully infused. In order to separate the flow media and fiber reinforcement, peel ply, a heat scoured and set polyester with a release agent coating, was placed on top of the flow media and overlain by the fiber reinforcement. The peel ply prevents the flow media from bonding to the fiber reinforcement. The fiber reinforcement used for the sandwich panels, as described in the panel specifications, was E-glass woven roving fabric. Either one or two layers were placed on
14
each side of the panel's core depending on the core material type. The core material was placed on top of the bottom layers of fiber reinforcement and then sandwiched by the final top layers of fabric. Mirroring the bottom layers, peel ply was placed over the fiber reinforcement with the upper flow media placed on top. The final layer was the vacuum bag. However, before the vacuum bag was placed, the resin inlets and vacuum distribution media were installed. Each panel has one resin feed line that allows resin to flow into the part, while both panels share a single polyester bleeder mat that spreads the vacuum over the entire width of the panel. A typical panel fabrication layout is illustrated in Figure 2.2. An image of the actual panel fabrication layout with the vacuum bag attached is depicted in Figure 2.3.
15
Figure 2.3 Panel Fabrication Layout This particular layout was used to fabricate two panels at once. As shown in the Figure 2.3, two resin inlets were used on the outside of the fabrication setup with the polyester bleeder in the center. The resin inlets extended the entire width of the panel to create a uniform flow front. Each inlet consisted of a wire coil with a diameter of 12.5 mm and was placed across the width of the panels. The coil was attached to a section of reinforced vinyl tubing, which led to a resin reservoir, which fed each inlet. It was important that the resin reservoir was placed at a lower elevation than the fabrication area to eliminate any siphoning effects that could have taken place during the infusion process. The polyester bleeder was placed across the entire width of the panels and in the center of the setup to create an even vacuum front on the opposite end of the inlets. The vacuum bag, which is the last layer, was cut much larger than the actual fabrication area, typically 1 Vi times larger. The bag was oversized to allow for wrinkles to be formed at each corner of the panel helping to release tension on the bag at those locations. The vacuum bag was attached to the tacky tape starting at one corner and
16
working around the part. It was very important to create a wrinkle at each corner of the panel to prevent the membrane tension from developing in the bag. Wrinkles were formed by folding a portion of the vacuum bag over a short piece of tacky tape. Two of the wrinkles were formed around the vinyl tubing used for the resin inlets, where tacky tape must be wrapped around the tubing and attached to the surface tape, as well as the vacuum bag. A small hole was cut in the center of the bag to allow one end of reinforced vinyl tubing to touch the polyester bleeder with the other end connected to a vacuum pump. Tacky tape was wrapped around the intersection of the bag and tubing to prevent any air leaks. Once the vacuum bag was sealed around the entire fabrication area the inlet tubes were clamped and the vacuum pump was turned on. All leaks were located and sealed thus ensuring that the part was maintaining the appropriate vacuum level of one atmosphere. An ultrasonic leak detector was used to locate the smaller leaks. 2.4.2. Infusion Process Once full vacuum was achieved, the resin was mixed. In order to minimize voids in the panel, it was necessary to mix a resin such that the gel time occurred 2 to 3 minutes after wetting out the panel. The selected elastomer modified vinyl ester epoxy resin, Dow Derakane 8084, was promoted with a 6% cobalt solution and accelerated with Dimethylaniline (DMA). A non-foaming Cumene hydroperoxide, Trigonox 239A, from Akzo Chemicals was used to catalyze the resin. Trigonox 23 9A was selected over Methyl Ethyl Ketone Peroxide (MEKP) to reduce the resin off gassing typically experienced with MEKP peroxides. Therefore, the resin could be infused directly after being mixed, unlike the normal MEKP, which had to sit for 5 to 10 minutes after mixing to allow for off-gassing.
17
Approximately 10 kg of vinyl ester was pumped into a resin reservoir to infuse the two 610 mm x 1220 mm panels. It is important to note that the amount of resin contained in the final part is significantly less than the amount required for fabrication. The flow media and resin distribution lines are typically filled with resin and additional resin is needed to ensure that the resin supply is not exhausted before the resin gels. The proper amounts of cobalt, DMA, and catalyst were mixed into the resin in accordance with Table 2.4, where each chemical was mixed in by percent of the total resin weight. Table 2.4 Typical Gel Times for Resin Mixtures (Dow Chemical)
15°-20°C Cool 60s°F
21°-26°C Mild 70s °F
27°-32°C Warm 80s °F
Peroxide
3.00%
3.00%
3.00%
Cobalt
0.60%
0.50%
0.40%
DMA
0.40%
0.30%
0.20%
Peroxide
2.00%
2.00%
2.00%
Cobalt
0.60%
0.50%
0.30%
DMA
0.30%
0.20%
0.20%
Peroxide
2.00%
2.00%
1.50%
Cobalt
0.60%
0.50%
0.30%
2.4.3. Infusion After the resin was properly mixed, the free ends of the resin inlets were placed in the resin reservoir. With full vacuum applied to the part, the inlet tubes were undamped to allow free flow of resin into the panels. Quickly, a flow front developed and traveled
18
across the entire panel as shown in Figure 2.4. Once both flow fronts reached the polyester bleeder, in the middle of the two panels, vacuum was reduced from one atmosphere to lA an atmosphere to prevent removal of styrene vapor from the resin and to minimize the amount of resin pulled into the resin traps. As soon as the excess resin in the supply bucket gelled the resin inlets were clamped and the panels were left to cure overnight at 24° C. Once cured, the flow media and peal ply were removed revealing the final part. It was not deemed necessary to post cure panels since an ambient cured matrix was strong enough to withstand the design loads.
Figure 2.4 Resin Flow Front
2.5. Conclusions and Recommendations The VARTM/S CRIMP method was successfully implemented to fabricate four sandwich composite panels, as presented previously in Chapter 2. Upon visual inspection, each sandwich composite panel exhibited high quality, showing no noticeable voids or under infused fibers. Since the E-glass fibers became transparent upon infusion, 19
the bond line between the face sheets and the core material could be visually inspected and showed excellent quality. Additionally, by creating a panel with the core material visible, the core failure modes became more apparent when characterizing the failure modes of each panel during the plate bending using the hydromat test system. The core shear failure modes were completely visible for these panels. Conversely, the carbon fiber reinforced sandwich composite panels that are tested in Chapter 3, completely hide the core material failure modes. The presented fabrication method can be expanded to sandwich composite panels and structures with a variety of cores and fiber reinforcements. Additionally, the fabrication method can be scaled, thus enabling the fabrication of large parts. The primary scaling limitations of the VARTM/SCRIMP method are the vertical height for fabricating large parts (pressure head issues) and the empirical fabrication rules learned by trial and error.
2.6 References Cited Bertelsen, W. D. (1992). Hydromat system. An experimental technique for the static and fatigue testing of sandwich panels. 2nd International Conference on Sandwich Constructions, Gainesville, Florida, EM AS. Bertelsen, W. D. (1994). Physical Testing of Marine Composites Using the Hydromat Test System. The Society of Naval Architects and Marine Engineers, Ann Arbor, MI. Bertelsen, W. D. (2000). Development and Certification of the ASTM D 6414-99 Plate Flexure Test and Its Implications for the Future of Composite Panel Design and Construction, Bay City, MI, Gougeon Brothers. Smith, S. A. (2001). Vacuum Assisted Resin Transfer Molding of Sandwich Structures: Material Processing, Evaluation, Fracture Testing and Analysis. Mechanical Engineering. Greensboro, North Carolina A&T State University: 323.
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3. IMPLEMENTATION OF A PLATE BENDING TEST SETUP UNDER UNIFORM LOAD HYDROMAT TEST SYSTEM 3.1. Abstract Sandwich composite panels are made from high stiffness face sheets and lowdensity core materials. The use of sandwich composite panels is becoming increasingly widespread in transportation and marine structures due to their high strength to weight ratio and their corrosion resistance. With the growing market for sandwich panels, new tests are needed to characterize their structural performance. The hydromat test system, ASTM D6416, is a recent experimental approach to characterize sandwich composite panels. This chapter introduces the hydromat test system (HTS) and illustrates how the HTS can be used to determine the structural performance of sandwich composite panels. 3.2. Introduction The use of sandwich composites, a material system consisting of a high stiffness and high strength face sheet sandwiching a low-density core, is a continuing trend in structural engineering. Sandwich composite materials provide a stiff, corrosion resistant part with high strength and stiffness when compared to more conventional structural materials. Currently sandwich construction is becoming more common in marine structures, especially in boat hulls. With the increased use of sandwich construction for marine applications there has been an increase in sandwich composite part failures. The hydromat test system was developed to reproduce the failures that sandwich panels were experiencing while in service (DeSautel and Sikarskie 1997).
21
The hydromat test system (HTS) was developed by William Bertelsen from Gougeon Brothers, Inc. through a partnership with Michigan Technological University (Bertelsen 1992; Bertelsen 1994; Rau, Bertelsen et al. 1994; Sikarskie, Eyre et al. 1995; DeSautel and Sikarskie 1997; Ahtonen and Sikarskie 1998; Bertelsen 2000). The HTS system was designed to simulate the hydrostatic loading configuration typically found in marine structures. The hydromat test system loads a panel with a two-dimensional distributed load, which is more realistic than previous approximations using onedimensional beams and concentrated loads to simulate actual ship hull loading. It has been shown that the hydromat test system allows repeatable experiments with results very close to predictions based on plate models (Ahtonen and Sikarskie 1998). A primary objective of this chapter is to understand important experimental issues with HTS and to propose solutions consistent with characterizing both stiffness and strength of sandwich composite panels. 3.3. Test Setup The HTS consists of an aluminum test fixture mounted on a servo-hydraulic test frame, as shown in Figure 3.1.
22
Figure 3.1 Hydromat Test System A water filled pressure bladder (see Figure 3.2), supported by a reinforced glulam support slab, was positioned directly below the fixture. The bladder support slab was bolted to a 100 kN load cell which was bolted to a t-slotted table.
Figure 3.2 500 mm Hydromat Pressure Bladder
23
The test fixture was machined from 6061-T6 aluminum according to the adjunct for ASTM D6416 standard test procedure. It consists of upper and lower panel support frames, as shown in Figure 3.3. Load From Test Frame
Figure 3.3 Hydromat Test Frame (lower support frame omitted) Test panels were equipped with line load diffuser strips, as shown in Figure 3.4 and Figure 3.5. The purpose of the line load diffuser strips was to distribute the line loads from the support frame journals to avoid core crushing and face sheet damage. Each panel tested was placed in the lower panel support frame, which was bolted to the upper panel support frame with 4 corner bolts. The proper corner bolt torque was determined on a panel to panel basis according to ASTM D6416. The hydromat bladders, used in this research, were constructed with either 2-ply or 3-ply industrial belting housed by a metal frame as shown in Figure 3.2. Each bladder was filled with water and completely closed, sealing the water inside the bladder. The maximum pressure allowed by the full
24
size 500 mm 3 ply bladder was 0.90 MPa, while the reduced sized 2 ply 305 mm pressure bladder could sustain 0.65 MPa. A pressure transducer (Setra 206) with a maximum capacity of 0.69 MPa was attached to each bladder for all the tests presented in this thesis. -5085mm
'Lire Load Diffuser Strips
5085mm
I
I
Figure 3.4 Line Load Diffuser Strips (Plan View) Brass Support Journals Line Load Diffuser Strips
Face Sheets
Core Material
Figure 3.5 Sandwich Plate Section and Edge Support Detail
25
3.4. Hydromat Modifications 3.4.1. Small Bladder According to ASTM, the size of the hydromat bladder should directly correspond to the size of the hydromat test system's support frame (ASTM 2001). As a modification to the test, a smaller bladder with dimensions of 305 mm x 305 mm was substituted for the 500 mm x 500 mm hydromat bladder, which was typically used in conjunction with the 500 mm support span test fixture. There were two advantages of using a smaller hydromat bladder with the 500 mm support span fixture. The first advantage was that the smaller bladder allowed for larger panel deflections without interference with the hydromat fixture. When using a full size bladder, the lower panel support frame can interfere with the pressure transducer when deflections are over 20 mm. The smaller hydromat bladder eliminates possible interference with the lower panel support frame, since the pressure transducer was further away from the lower panel support frame. The second advantage was that the maximum core shear stress occured further away from the panel support span. Thus, it was more probable to have a core shear failure that was not influenced by the panel supports, although failures with the standard size bladder were typically found away from the supports as well. 3.4.2. Dynamic Loading Inertial Compensation When the hydromat test system was applied to dynamic testing it was important to consider where the load cell was mounted. In previous work, the hydromat test system fixture was a stationary component and the hydromat pressure bladder was pushed by a servo hydraulic ram into a fixed panel (ASTM 2001). In this test setup, the load cell was
26
always stationary since it was placed above the test fixture. Thus, a dynamic loading cycle would not create any inertial load affecting the load cell readings. In the experimental work presented in this thesis, the servo hydraulic test machine was inverted when compared to previous hydromat test system setups. Thus, the servo hydraulic ram loaded the fixture from the top while the pressure bladder remained fixed, as shown in Figure 3.1. To prevent inertial effects during dynamic loading, the load cell must be attached to the stationary component of the HTS. Thus, the load cell was mounted under the bladder support slab instead of on top of the hydromat test fixture as done in the previous work. There were two advantages of the dynamic loading inertial compensation. The first advantage was that the load measured by the load cell was accurate. If the load cell was reading dynamics effects due to accelerations applied to the test frame, the true load applied to the panel would be unknown. The second advantage was that the Proportional, Integral, Derivative (PID) tuning for the control loop on the servo hydraulic test machine was more manageable. When the load cell was reading inertial effects from the test frame, the control loop for the servo hydraulic test machine would become unstable and exhibit an undesirable resonance. 3.4.3. 3-D Digital Image Correlation An improvement was made for the instrumentation of the hydromat test system. A 3-D digital image correlation (IC) system was mounted to the hydromat test fixture. The location of the two digital CCD cameras is shown in Figure 3.1. The 3-D IC system allowed for full field measurements of the in-plane strain components and Zdisplacements of a hydromat panel. By integrating the IC system with the HTS system,
27
the traditional instrumentation to measure strains and displacements can be replaced. The IC system was expected to record more reliable and complete data relating to a hydromat test panel's response. 3.5. Sandwich Panel Response Verification The hydromat test system was used to verify the response of two sandwich composite panels. Each panel has a carbon fiber cross-ply laminate face sheet with either an end-grain balsa core or a polyetherimide foam core. Constituent properties of the sandwich panels, face sheets and cores, were quantified using standard ASTM tests. Using data from the material characterization, the two sandwich panels were designed to exhibit core shear failure as the dominant failure mode. Each panel was fabricated using the VARTM/SCRIMP process. The HTS was used to evaluate the structural response of the panels. Data from the hydromat test was compared to predictions from the analytical sandwich composite plate solution presented in Chapter Four. 3.5.1. Carbon Composite Properties Physical and mechanical properties of the carbon fiber face sheets were characterized. The ultimate tensile strength in the fiber direction, elastic modulus and Poisson's ratio were determined using the standard tensile test, ASTM D3039 (ASTM 2000c). The ultimate tensile strength and elastic modulus in the transverse direction were determined using the ASTM D638 test standard (ASTM 2002). ASTM D638 was selected for the transverse tensile testing due to the dog bone specimen shape. It was thought that the rectangular specimen of ASTM D3039 would exhibit a higher percent of grip failures than the dog bone shape used in ASTM D638. Although a study was not performed, ASTM D638 produced acceptable failure modes for all of the test coupons.
28
Additionally, ASTM D638 does not require tabbing of the specimens where ASTM D3039 may require tabs to prevent grip failures. It should be noted that ASTM D638 is not recommended for tensile coupons with an axial modulus over 20 GPa. The ultimate compressive strength in the two material directions were determined using the ASTM D3410 test standard test (ASTM 1995). The in-plane shear modulus was determined using ASTM D5379 V-notched beam shear test (ASTM 1998). Mean values of each property as well as the coefficient of variation (COV) are reported in Table 3.1 Table 3.1 Mechanical Properties of Carbon Fiber Composite Sheets Mean
COV %
319.4
8.1
108.4
7
1817
0.8
17.3
8.6
120.3
4.4
7.5
7.3
4.38
6.7
0.315
9.9
Experimental Property 0° Ultimate Compressive Stress, MPa 90° Ultimate Compressive Stress MPa 0° Ultimate Tensile Stress MPa 90° Ultimate Tensile Stress MPa 0° Modulus of Elasticity GPa 90° Modulus of Elasticity MPa Shear Modulus Poisson's Ratio
3.5.2. Core Shear Properties The core shear properties for the balsa and foam cores were determined using ASTM C273 (ASTM 2000a). The test fixture shears a sample of core material between two steel plates. The steel plates were sand blasted to maximize the mechanical bonding
29
between the plates and the sample. Each core material was bonded to the steel plate using a highly toughened epoxy structural adhesive (ProSet 276) and hardener (ProSet 176) from Gougeon Brothers. This structural adhesive was specifically designed to resist peeling stress and exhibits a high shear strength compared to other adhesives. The test samples were clamped using a moderate clamping pressure and allowed to cure for at least 24 hours. Twelve foam samples and twelve balsa samples were tested. The mean value of sample density, ultimate shear strength and shear modulus of elasticity are reported in
Table 3.3 with the published data from the manufacturer presented in Table 3.2. It should be noted that the manufactures strength properties for the S45 shear strength were specified for a 12.7 mm ( Vi-in) balsa core, where this study tested a 25.4 mm (1-in) balsa core. Previous work has shown that the shear strength of balsa wood decreases as the core becomes thicker due to cell instabilities, similar to the instabilities found in honeycomb cores. Thus for the 25.4 mm (1-in) thick end grain balsa core the shear stiffness is expected to be 14% lower than the published manufacture's values (McGeorge and Haynian 1998). Table 3.2 Manufacturer's Properties for Lightweight Core Materials Property
End-Grain Balsa Wood
Structural Foam Core
Density, kg/mJ
82
Ultimate Shear Strength, MPa
1.6
1.1
Shear Modulus, Mpa
97
23
30
~
80
Table 3.3 Computed Core Material Properties (ASTM D638) End-Grain Balsa Wood Experimental Property
Structural Foam Core
Mean
COV %
Mean
COV %
J
76.9
~
79.0
—
Ultimate Shear Strength, MPa
1.4
7.3
0.98
2.6
104.5
17.4
23.7
5.2
Density, kg/m
Shear Modulus, MPa
3.5.3. Sandwich Panel Fabrication The sandwich composite test panels were fabricated using the vacuum-assisted resin transfer molding (VARTM) process with the proprietary SCRIMP technology. Each sandwich composite panel face sheet was constructed from unidirectional carbon fibers with a vinyl ester matrix. The carbon fiber was supplied by JB Martin and had a weight of 408 g/m2 (11 oz/yd2). Each face sheet consisted of four layers of fabric reinforcement in a [0,90,0,90] stacking sequence. Then, the resulting sandwich panel had balanced and symmetric face sheets. The polymer resin was Derakane 8084, which is an elastomer modified epoxy vinyl ester resin. One test panel was fabricated with an endgrain balsa wood core. The core style was Baltek S45 rigid sheet with AL600/10 coating for decreasing porosity and increasing bond strength. The other panel was fabricated with a Baltek R82 polyetherimide foam core. To prevent excessive resin absorption during the resin infusion process the foam core was pre-coated with a vinyl ester resin film. The panel fabrication was conducted at ambient conditions with no post-curing process. In the VARTM/SCRIMP method the face sheets of the sandwich composite panels were infused and bonded to the core material in one step. Two panels with dimensions 610 by 610 mm were fabricated. Each fabricated panel was cut into a square test panel with dimensions of 508.5 mm using waterjet machining equipment. The edges
31
of the foam core panel were reinforced with the S45 balsa wood to prevent core crushing as required by ASTM D6416. 3.5.4. Modified Hydromat Testing The hydromat test panels were evaluated using a modified hydromat test system. The test follows the ASTM standard except that the standard bladder was replaced with a smaller (305 mm x 305 mm) pressure bladder. The reason for using this modified test setup was to allow larger deformations of the panel specimens without physical interference with the fixture supports. The maximum deflection of each panel was measured using a linear variable differential transformer (LVDT) located at the center of the panel. Bonded resistance strain gages on the top face sheet in both the x- and y-direction were placed at the panel center, adjacent to the LVDT tip. Pressure data from the water-filled bladder was monitored using a Setra 206 pressure transducer. The applied load from the servo hydraulic test frame measured using an Instron 100 kN load cell. 3.5.5. Modeling Predictions versus Experimental Results Maximum panel deflections and in-plane strain components were predicted based on orthotropic sandwich plate theory. The correlation between the predicted values and experimental values is presented in Table 3.4.
32
Table 3.4 Sandwich Plate Modeling Predictions versus Experimental Values
Experimental Z-Displacement (mm) Predicted Z-Displacement (mm) Z-Displacement Percent Error Experimental Strain X (u) Predicted Strain X (u) Strain X Percent Error Experimental Strain Y (u) Predicted Strain Y (u) Strain Y Percent Error
Carbon Balsa Carbon Foam 6.667 3.47 3.41 7.134 6.5% 1.8% 1512 1379 1334 1334 3.4% 13.3% 1466 1336 1316 1316 11.4% 1.5%
All strain quantities for the carbon balsa test panels showed an excellent correlation to sandwich plate theory. The carbon foam test panel exhibited a deviation from the sandwich plate modeling. The exact cause of the deviation is unknown but it may relate to the fabrication process. During fabrication, the foam core panel absorbed resin which can not be accounted for in the modeling. As the core absorbed resin, the core should stiffen in both bending and shear. The load versus strain curves for the carbon balsa and carbon foam panels are illustrated in Figure 3.6 and Figure 3.7, respectively. It is worth noticing that both plots exhibit a very linear response with the carbon foam panel exhibiting a slight membrane stiffening effect at larger deformations. The sharp drop in strain illustrated in Figure 3.6 was caused by the core of the carbon balsa panel failing in shear. Once the core failed, the stress was relieved from the face sheets.
33
500
3000
1500 Microstrain
Figure 3.6 Carbon Balsa Load versus Strain X
1000
1500
2000
2500
3500
4000
Strain X (Microstrain)
Figure 3.7 Carbon Foam Load versus Strain X The load versus transverse displacement curves for the carbon balsa and carbon foam panels is illustrated in Figure 3.8 and Figure 3.9, respectively. The carbon balsa panel exhibits a linear behavior until failure, while the carbon foam panel shows a significant non linearity. This non linearity developed at higher loads where the core material was yielding locally. The orthotropic sandwich plate theory used for the hydromat modeling
34
cannot account for the non linear shear properties of the foam. The model assumes constant shear stiffness throughout the test. A prior model derived by the Michigan Technological University (MTU) research team can account for some bilinear behavior in the core shear modulus as long as the core does not become perfectly plastic. However, this model cannot accurately represent the elastic plastic deformation that occurs in the R82 core (Sikarskie and Mercado 1999).
25
|20 •a
8 -• 15
Transverse Displacement (mm)
Figure 3.8 Carbon Balsa Load versus Transverse Displacement
35
10
15
Transverse Displacement (mm)
Figure 3.9 Carbon Foam Load versus Transverse Displacement 3.5.6. Experimental Failure Modes The failure modes of the balsa panels correlated to the failure modes of the ASTM C273 test specimens as shown in Figure 3.10 and Figure 3.11, indicating that the panels exhibited core shear failure. The foam panels in the hydromat tests, as expected, exhibited the same failure modes as the foam core shear specimens. However, the panels could not be loaded to a shear strain level high enough to tear the core. Upon inspection, the foam cored panels exhibited a definite plastic deformation leaving the test panel with permanent shear deformation.
36
Figure 3.10 Typical Balsa Shear Failure Mode in ASTM C273
Figure 3.11 Core Shear Failure Mode of the Balsa Carbon Panel With regards to the balsa wood sample, the model was successful in determining shear failure in the panel core. The final load before failure in the carbon balsa panel was 30.9 kN with a predicted maximum core shear stress of approximately 1.44 MPa, a stress sufficient to create shear failure in the sample according to the ultimate shear strength of the balsa wood determined by the ASTM C273 test. The model also predicts the location of maximum shear stress. The maximum core shear stresses were predicted to occur along centerlines of the panel 135 mm from the edges support as shown in the contour plot in Figure 3.12.
37
Figure 3.12 Typical Core Transverse Shear Stress Distributions ixz and xyz The predicted shear stress distribution on a centerline of the panel is shown in Figure 3.13. The location of maximum shear stress occurs at the edge of the hydromat bladder footprint. After testing, the samples were cut in quarters and examined. A balsa wood shear failure was discovered approximately 114 mm from the edge support, indicating that the model was able to predict with some acceptable accuracy the location of the failure.
38
0 1Su
50 PP°rt
100
150 Distance from Support (mm)
200
250 Panel Center
Figure 3.13 Predicted Core Shear Stress Distribution for the Carbon Balsa Panel The foam core panel was more complicated to model. Since no catastrophic failure occurred, the actual failure load could not be quantified. The sandwich composite plate model predicted that failure would initiate at 22 kN based on the experimental core shear strength. A definite non linearity developed in the load versus deflection curve for loads above the 22 kN threshold as seen in Figure 3.9. The sandwich plate model predictions deviate from the actual panel response once core yielding occurs. The carbon foam panel was loaded up to 35.6 kN. The model, assuming a constant shear modulus and no yielding, predicted a core shear stress associated with the maximum load of 1.6 MPa, which is 60% more than the ultimate shear strength of the material. On a design note, the plastic failure mode observed in the foam panel would be desirable when a ductile failure mode is needed for safety.
39
3.6. Conclusions and Recommendations The hydromat test system proved to be a versatile method for testing sandwich composite panels. Necessary changes in the hydromat test setup, such as changing bladder sizes or load cell positioning, were made with relative ease. Sandwich plate modeling was used in conjunction with the hydromat test system to verify the structural response of a sandwich panel. This combined experimental and numerical technique opens the door for inverse solution methods that allow back computation of sandwich panel stiffness properties. The sandwich plate model was accurate in predicting the level of load and pressure corresponding to yielding of ductile foam cores or fracture of brittle balsa wood cores, as presented in Chapter 3. The model predicted the failure location for both materials; however, the model could not predict the ultimate failure load for the foam panel since the core yielded and carried a significant amount of load above the predicted failure load. A non linear material model would be needed to accurately model the yielding response of the foam core. In summary, accurate modeling and material characterization are essential when designing sandwich composites for critical structures. The hydromat test system allows a designer to verify the structural response of a sandwich panel, ensuring the panel performs in a predictable manner. It was shown that that a significant amount of testing was required to characterize the response of a sandwich composite panel. Findings presented in Chapters 5 and 7 add to the hydromat test system capabilities by investigating inverse solution methods, which have the capability of quantifying the
40
sandwich panel stiffness with a smaller testing effort. The method combined 3-D digital image correlation with the hydromat test system and sandwich plate modeling. 3.7. References Cited Ahtonen, P. H. and D. Sikarskie (1998). "An Analytical and Experimental Comparision of Orthotropic Sandwich Panels using the Hydromat Test System." Composites Part B29B: 705-714. ASTM (1995). D3410 Standard Test Method for Compressive Properties of Polymer Matrix Composite Materials with Unsupported Gage Section. West Conshohocken, PA, American Society of Testing and Materials. ASTM (1998). D5379 Standard Test Method for Shear Properties of Composite Materials by the V-Notched Beam Method. West Conshohocken, PA, American Society of Testing and Materials. ASTM (2000a). C273 Standard Test Method for Shear Properties of Sandwich Core Materials. West Conshohocken, PA, American Society of Testing and Materials. ASTM (2000c). D3039 Standard Test Method for Tensile Properties of Polymer Matrix Composite Materials. West Conshohocken, PA, American Society of Testing and Materials. ASTM (2001). D6416 Standard Test Method for Two-Dimensional Flexural Properties of Simply Supported Sandwich Composite Plates Subjected to a Distributed Load. West Conshohocken, PA, American Society of Testing and Materials. ASTM (2002). D638 Standard Test Method for Tensile Properties of Plastics. West Conshohocken, PA, American Society of Testing and Materials. Bertelsen, W. D. (1992). Hydromat system. An experimental technique for the static and fatigue testing of sandwich panels. 2nd International Conference on Sandwich Constructions, Gainesville, Florida, EMAS. Bertelsen, W. D. (1994). Physical Testing of Marine Composites Using the Hydromat Test System. The Society of Naval Architects and Marine Engineers, Ann Arbor, Michigan. Bertelsen, W. D. (2000). Development and Certification of the ASTM D 6414-99 Plate Flexure Test and Its Implications for the Future of Composite Panel Design and Construction. Bay City, Bougeon Brothers, INC. DeSautel, J. M. and D. L. Sikarskie (1997). On Defining Core Properties for Composite Sandwich Panels. Mechanical Engineering. Michigan, Michigan Tech. McGeorge, D. and B. Haynian (1998). Shear Strength of Balsa-Cored Sandwich Panels. Sandwich Constructions, Stockholm, Sweden, EMAS. Rau, C. S., W. D. Bertelsen, et al. (1994). On the Development of a Two-Dimensional Test Fixture for Composite Panels. Fifth International Conference on Marine Applications of Composite Materials, Melbourne FL., Composite Education Association Inc. Sikarskie, D. L., M. Eyre, et al. (1995). Verification of the Hydromat Test System as a Viable Means of Testing Two-Dimensional Sandwich Panels. Third Int. Conference on Sandwich Construction, University of Southampton, United Kingdom.
41
Sikarskie, D. L. and L. L. Mercado (1999). "On the Response of a Sandwich Panel with a Bilinear Core." Mechanics of Composite Materials and Structures 6(1): 57-67.
42
4. VALIDATION OF AN ORTHOTROPIC SANDWICH COMPOSITE PANEL MODEL 4.1. Abstract This chapter provides an overview of classical lamination theory, first order shear deformation theory and sandwich plate theory as it applies to hydromat test panels. The chapter presents the complete methodology used to predict the response of a hydromat test panel and foreshadows future inverse solution methods. In previous work, an orthotropic sandwich composite plate model was developed and used to model a hydromat test panel. Upon inspection of the previous model, it was apparent that the solution was oversimplified. This chapter illustrates a formal derivation of an improved orthotropic sandwich composite plate model, which accurately represents the hydromat test system. In this chapter, the derived model was compared to the previous hydromat modeling and finite element modeling (FEM) results. 4.2. Hydromat Testing Introduction The hydromat test system was developed by William Bertelsen from Gougeon Brothers, Inc. to provide the marine industry with a test that simulates the loading of sandwich panels under hydrostatic pressure (Bertelsen 1992; Bertelsen 1994; Bertelsen 2000). The ASTM D30 Committee on Composite Materials reviewed the hydromat test and adopted as a standard test method in 1999. The standard is designated as ASTM D6416 "Standard Test Method for Two-Dimensional Flexural Properties of Simply Supported Plates Subjected to a Distributed Load" (ASTM 2001). The test procedure is performed using quasi static loading to determine both the performance response and the stiffness parameters of a sandwich composite panel, as follows: deflection versus load,
43
face sheet strain versus load, bending stiffness parameters, shear stiffness parameters, ultimate strength, and failure modes. To determine the bending and shear stiffness parameters of a sandwich composite panel, ASTM D6416 considers an inverse solution method. The inverse solution method is based on an isotropic sandwich plate solution, thus the solution is only valid for sandwich composite panels with isotropic face sheets and isotropic cores. The hydromat test system (HTS) consists of two parts, a support frame and a pressure bladder. The frame, illustrated in Figure 4.1, creates a simply supported boundary condition around the edges of a test panel while the pressure bladder, or hydromat, applies a uniform distributed load to the panel. A plan view of a typical test panel with the applied hydromat loading is shown in Figure 4.2. A detail of the journals that create the simple supports on the panel edges is depicted in Figure 4.3. Load From Test Frame
Pressure is Applied to the Bottom of the Test Panel
Figure 4.1 Hydromat Test Frame (lower support frame omitted) 44
1
/Pressure Palch
0
— 0—
0
Supports
i
y
Q
Figure 4.2 Support Span and Pressure Patch Brass Support Journals Line Load D iff user Strips
Face Sheets
Core Material
WSF* \ \ \ \ \ \
Figure 4.3 Sandwich Plate Section and Edge Support Detail A key advantage of the hydromat test configuration is the similarity to Navier's solution for simply supported plates under uniform loading. Using Navier's plate solution, the response of a test panel can be predicted and then verified experimentally with the HTS. It should be noted that the Navier's solution form adopted for modeling 45
_
simply supported plates is valid for: isotropic plates, isotropic sandwich plates, specially orthotopic plates, and specially orthotropic sandwich plates. Therefore, the solution form is valid provided bending-extension coupling does not develop. The focus of this chapter is to formally present a comprehensive orthotropic sandwich composite model that accurately represents the hydromat test system. 4.3. Sandwich Plate Overview Sandwich structures are composed of two primary constituents, the facings or face sheets and the core material. Typically, the facings of a sandwich structure are thin membrane-like materials with a very high stiffness, while the core material, being quite thick compared to the face sheets, is composed of a light weigh and relatively compliant material. The primary function of the face sheet is to carry tension and compression from axial loading and bending moments applied to the structure. The function of the core in the sandwich structure is to stabilize the face sheets, preventing premature local buckling, and at the same time providing separation between the face sheets, which greatly increases the bending stiffness of the structure (i.e., I-beam effect). Additionally, the core is expected to carry the entire transverse shear force applied to the structure. 4.4. Analytical Solution The face sheets of a general orthotropic panel for hydromat tests are constructed from orthotropic composite laminas in a stacking sequence that creates a balanced and symmetric sandwich structure. A symmetric sandwich panel may exhibit orthotropic behavior but does not exhibit bending-extension coupling (By =0), which is found in an unsymmetric structure. Furthermore, a balanced sandwich panel will have zero values for the in-plane shear-extension stiffness coupling coefficients (A )6 = A 26 = 0). It is
46
important to note that the torsional-bending coupling coefficients Di 6 and D26 may have a non-zero value even if the panel is symmetric and balanced. A specially orthotropic laminate represents the case where the torsional-bending coupling coefficients are equal to zero. Violation of the specially orthotropic condition can be quantified using a non dimensional measurement presented in equation (4.1). The violation of the specially orthotropic condition is minimized when a balanced and symmetric orthotropic sandwich composite plate has thin face sheets. Each composite lamina in the face sheet is defined in a material coordinate system with axes labeled as 1, 2, and 3 as defined in Figure 4.4. If the fiber reinforcement is a woven roving, then the 1 direction corresponds to the warp and the 2 direction corresponds to the fill. ^ 3 - Direction (Corresponds to Z) 2 - Direction (Transverse to Fibers)
1 - Direction (Parallel to Fibers)
Figure 4.4 Lamina Coordinate System Typically, a structure will have a global coordinate system that differs from the materials coordinate system of each lamina. The angle, 9k between the x-axis of the global coordinate system and the 1-direction of the composite lamina coordinate system defines the orientation of the lamina. Once the laminas are properly stacked, creating the face
47
sheets, an orthotropic core is bonded between the two face sheets. A typical sandwich structure resulting from this assembly is represented in Figure 4.3.
Figure 4.5 Sandwich Panel Section with z-Coordinates The described sandwich structure with orthotropic face sheets and an orthotropic core is a general analytical sandwich composite solution applicable to the hydromat test system. This analytical approach was selected because orthotropic sandwich plate theory provides a robust solution that models a variety of sandwich plates ranging from highly orthotropic sandwich plates to sandwich plates constructed from total isotropic materials. For example, if the core was isotropic instead of orthotropic, the two shear modulus terms defining the core would be set equal, thus reducing the solution to the case of isotropic
48
shear deformations. Furthermore, the modeling can be used to represent laminates that violate the specially orthotropic constraints relating to the coupling terms. Equation (4.1) is a non dimensional measure of the how close a laminate, with D ]6 and D26 ? 0, is represented by an equivalent specially orthotropic material model under a bending load (Barbero 1998). Additional non dimensional measures are available to quantify the approximation error associated with non zero By, A ]6 and A26 terms (Barbero 1998). Finite element modeling can also be implemented to model these more complex laminates.
(D
]
2
I A. J
+
(D }
The presented modeling for the orthotropic sandwich composite panel was organized into two parts: stiffness determination and structural behavior. The stiffness determination relates to section properties and is completely independent of how the sandwich structure is loaded. For example the bending stiffness and shear stiffness parameters of beam under 4-point bending is exactly the same as the bending and shear stiffness parameters of a plate under uniform loading. By applying classical lamination theory and first-order shear deformation theory (FSDT) the bending and shear stiffness parameters of a section are quantified. Then, displacements and strains resulting from a hydromat test can be computed using sandwich plate theory.
4.4.1. Classical Lamination Theory As a starting place for modeling, classical lamination theory based on the textbooks by Hyer and by Whitney was adopted(Whitney 1987; Hyer 1997; Zhao 2002).
49
A practical first step in composite analysis is to determine the assumptions related to the behavior of the structure. The structure being modeled is a hydromat test panel with thin face sheets and a thick core. The face sheets are assumed to be in a state of plane stress based on equation (4.2). a
=x zz
(4.2)
=T =0 xi
yz
A plane stress assumption greatly simplifies the constitutive relations for each of the composite lamina that make up a face sheet laminate. Then, the stiffness matrix defining the composite lamina is reduced from a 6x6 matrix containing 9 separate terms to a 3x3 matrix of 4 separate terms as in equation (4.3). Each term of the reduced stiffness matrix is defined through the engineering constants defining the laminae in equation (4.4). Qu - n Qn 0
,
Qn =
Q22
0 *n 0 O £r22
0
Qee] Yn.
Qn n
n
\-vnv2i
-
v E
"i 1-^2^21
(4.3)
(4.4)
Q66=G
022 = 1-W21
The reduced stiffness matrix for each lamina is transformed from material coordinates to the global coordinates. With the angle 0k between the global coordinate system and the material coordinate system known, the stiffness matrix of each lamina can
50
be transformed into global coordinates as presented in equation (4.5). Each of the terms in the transformed stiffness matrix is found using (4.6). (4.5)
(4.6)
The stiffness of each lamina in the face sheet is defined in global coordinates. From this, the structural properties of the sandwich plate can be computed. The in-plane stiffness is defined through the A matrix, bending stiffness parameters are defined through the D matrix, and the B matrix represents the bending-extension coupling terms. The A, B, and D matrices are formed using (4.7) combined with layer's z-coordinates as shown in Figure 4.5. Together the A, B, and D matrices can be stacked and augmented to create the ABD matrix, which is used to form the constitutive relations in (4.8). If the sandwich plate is symmetric, the B matrix will contain all zero terms since there is no bending-extension coupling. (Besides if the sandwich plate is balanced A}6 = A26 =0.) In addition, for sandwich panels with thin and balanced face sheets it is possible to assume D « 0 and D » 0. One practical experimental consideration is that in the hydromat test system, the in-plane force resultants TV,, Ny, and Nxy are not applied to the panel and therefore are given a null value. 51
(4.7)
(4.8)
When transverse shear forces are present an additional set of constitutive relationships are needed. The core of the sandwich composite is assumed to carry all of the transverse shear forces applied to the sandwich structure. Equation (4.9) shows the constitutive relationships for the core with the A44 and A55 terms representing the shear stiffness of the sandwich plate as derived in (4.10). Typically for sandwich composite plates the shear correction factors IC44 and k55 are set to 1 (Whitney 1987) although they can be computed using energy methods. (4.9)
(4.10)
In summary, a set of equations resulting from the application of classical lamination theory to sandwich composite panels has been illustrated. The stiffness of each lamina was quantified and used to determine global stiffness matrices that define the sandwich section. Constitutive relationships based on these stiffness matrices, forces, strains and curvatures where shown. Using properties of the core material the shear
52
stiffness of the plate was quantified and a constitutive relationship between shear force and shear strain was illustrated. These sandwich plate properties combined with the proper governing equations, loading and boundary conditions can be used to accurately model the hydromat test system. 4.4.2. Sandwich Plate Solution as Applied to the Hydromat Test System As stated in the preceding section, the model developed in this chapter for hydromat test panels is specific to balanced and symmetric orthotropic sandwich composite plates with orthotropic cores. It should be noted that the following sandwich plate solution represents a specially orthotropic plate and can be used as an approximation for the response of a balanced and symmetric orthotropic plate. The modeling can also represent more general plates. For example, by equating specific parameters, panels with isotropic face sheets or isotropic cores can be modeled. The solution procedure begins with the three basic equations defining the deformations of a sandwich plate, (4.11), (4.12), and (4.13). A panel subjected to a hydromat test has mid-plane displacements u0 and v0 equal to zero, since no axial loads are applied. Thus, the in-plane displacements and in-plane strains are only functions of the panel rotations, i|/x and v|/y and the z-coordinate. u = u0+z-(//x
(4.11)
v = v0+z-y/y
(4.12)
w=w0
(4-13)
As stated earlier, a hydromat test panel has simply supported boundary conditions. These boundary conditions are described in (4.14) and (4.15). Additionally, 53
the behavior of the panel must conform to the relationships set forth in the governing differential equations (4.16), (4.18), and (4.20). Equations (4.16) and (4.18) can be simplified to Equations (4.17) and (4.19) respectively by taking into account the balanced and symmetric lay-up of the sandwich panel. (4.14)
=0
(4.15)
=0
(4.16)
(4.17)
(4.18)
(4.19) 0
54
Ga-h
di//x dx
d2w (dyy y d2w + G -h + q=0 +2i dx2 , By dy2 )
(4.20)
With the governing equations and boundary conditions defined, a solution form is needed that satisfies both the governing differential equations and the boundary conditions. The proper solution form was developed by Navier (Whitney 1987) and has the form of (4.22), (4.23), (4.24), (4.25). Notice that the series is summed over odd terms for this particular solution. Equation (4.22) has two experimentally determined parameters in the function, the pressure P and the distance ^ from the support to the pressure load. The hydromat bladder is assumed to distribute a square uniform pressure over the panel as depicted in Figure 4.2 (Rau, Bertelsen et al. 1994). The magnitude of the applied pressure can be determined by using a pressure transducer that measures the fluid pressure inside the hydromat. This pressure measurement when combined with a force measurement allows for the computation of from(4.26). With the solution in the proper form, the unknown constants Am„, Bmn and Cm„fromthe rotation equations and displacement equation can be solved by substitution into the 3 governing differential equations, (4.17), (4.19) and (4.20). Equations (4.27) through (4.36) represent the solutions to the constants Amn, Bmn and Cmn. (4.21)
(4.22)
(4.23)
55
(4.24)
(4.25)
(4.26)
(4.27)
(4.28) (4.29)
(4.30)
(4.31)
(4.32) (4.33) (4.34)
(4.35)
(4.36)
From the rotations and displacement equations the in-plane strains, ex, ey and yxy can be determined from (4.37), (4.38), and (4.39). Each one of the in-plane strain
56
equations is a linear function of the plate curvatures multiplied by a specific z-coordinate in the laminate. As stated previously, the in-plane strains are only functions of the panel rotations. Transverse shear strains in the core, yxz and yyZ; are determined from (4.40) and (4.41). With the strains in the sandwich panel defined, the constitutive relations presented in the classical lamination section are used to determine the stress distributions in the sandwich panel for the hydromat test. (4.37)
ox (4.38) £
>y=-z-^T.t/y
dy
Y» = ~z •
'' d_ dy
_d_ dx
(4.39) y
d
(4.40)
rX2=yx+—™ dx /yz
*
y
d dy
(4.41)
4.4.3. Previous Hydromat Orthotopic Sandwich Composite Plate Solution An orthotropic sandwich composite plate model was presented in previous work leading to the development of the hydromat test system. This plate model was developed by Michigan Technological University (MTU) and is based on Timoshenko plate theory. A complete description of the model can be found in (Ahtonen and Sikarskie 1998). The MTU orthotropic plate model differs from the modeling above in two main aspects: the bending and shear responses. The MTU solution is separated into a classical plate theory and a shear deformation theory. The classical plate theory is used to compute the 57
transverse displacements due to the bending and the in-plane strain components. The shear deformation theory is used to correct the transverse displacements by adding shear deformations to the bending response. Upon inspection of the solution, an over simplification is apparent. The solution neglects a coupling between the in plane strains and the shear deformation that occurs in most orthotropic plates. To illustrate the simplification, a finite element model will be created and used as a benchmark for comparison of the two solutions.
4.4.4. Finite Element Modeling Compared to Analytical Results A finite element method (FEM) model was constructed using the commercial finite element code ANSYS (ANSYS 2003). A description of the model is shown in Table 4.1 and Table 4.2. The hydromat panel is symmetric about two axes so a quarter model was created instead of a full model. The boundary conditions for the plate are simply supported as described by Reddy and Whitney (Reddy 1984; Whitney 1987). A quadratic laminated shell element with four corner nodes and four mid-side nodes called Shell 91 was adopted. This shell element was implemented to mesh the plate since it has a sandwich option, which follows sandwich plate theory. Table 4.1 FEM model Description Finite Element Model Parameters Element Type Degrees of Freedom Number Nodes Number of Elements
58
Value Shell 91 6 1976 625
Table 4.2 Physical Constants and Elastic Properties for Hydromat Test Modeling Geometric & Load
Values
Component
Material Properties
Values
a 0
500 mm 100 mm
Face Sheet Face Sheet
E, E2
140 GPa lOGPa
H
25 mm
Face Sheet
G]2
4 GPa
ti
1 mm
Face Sheet
V]2
0.25
P
222,222 Pa
Core
G13
50 GPa
F
20,000 N
Core
G23
50 GPa
A convergence study was performed to ensure that the mesh was refined to a level where the FEM solution would produce accurate results. The convergence of the maximum in-plane strain in the x-direction versus the number of elements is illustrated in Figure 4.6. Convergence Study (Quarter Model)
j
300
400
500
600
Number of Elements
Figure 4.6 FEM Convergence Study
59
700
800
900
The in-plane strain component was used for the convergence study since strain values are more sensitive to the convergence of the solution. For example, the maximum displacement value had converged to 4 significant figures for even the coarsest mesh of 25 elements even though the strain components had not converged to that level. Typically strain values converge slower because they are differentiated displacements and are much more sensitive to small changes in the displacement data. Two analytical solutions were compared to the finite element modeling. The first solution is based on the orthotropic sandwich composite plate theory (SW) developed in this chapter while the second solution corresponds to the orthotropic sandwich composite plate model developed by the Michigan Technological University (MTU) team (Ahtonen and Sikarskie 1998). The input parameters for the analytical solutions and the finite element model are shown in Table 4.2. In the series solution, 200 terms were considered in each analytical solution to ensure convergence. It is important to note that the in-plane strain components converge within about 25 terms while the transverse shear strains converge at a much slower rate for this particular orthotropic panel. The finite element model and the analytical solution based on the sandwich plate theory developed in this chapter are correlated in Table 4.3. Excellent agreement between the two solutions was observed. The MTU analytical solution was compared with the FEM solution in Table 4.4. In this case, it was found that the two solutions diverge. The error in the MTU solution was attributed to the simplifications taken in the definition of shear deformation in the MTU modeling. In the Whitney derived modeling,
60
a face sheet to core coupling exists that is neglected in the MTU approach. It is important to note that all the solutions show excellent agreement when: Ei = E2. Table 4.3 Analytical Sandwich Plate Solution versus Finite Element Modeling
Result
FEM Model
SW Solution
Percent Difference
Max In-Plane Strain x Direction
1201
1201
0.00%
Max In-Plane Strain y Direction
2805
2804
0.04%
Max In-Plane Strain xy Direction
3613
3612
0.03%
Max Core Shear Strain xz Direction
22707
22680
0.12%
Max Core Shear Strain yz Direction
7948
7930
0.23%
Max Panel Deflection (mm)
6.124
6.124
0.00%
Table 4.4 MTU Sandwich Plate Solution versus Finite Element Modeling
Result
FEM Model
MTU Solution
Percent Difference
Max In-Plane Strain x Direction
1201
1429
18.98%
Max In-Plane Strain y Direction
2805
1468
47.66%
Max In-Plane Strain xy Direction
3613
2554
29.31%
Max Panel Deflection (mm)
6.124
5.143
16.02%
4.5. Conclusions The presented Whitney derived sandwich composite plate modeling showed an improvement over previous hydromat plate modeling techniques. A comparison of both models to finite element modeling was presented. An excellent agreement existed
61
between the Whitney derived sandwich composite modeling and the finite element modeling. The previous modeling technique showed poor agreement to the F E M modeling. This error explains why some orthotropic sandwich plates tested in the literature deviated from previous modeling. As a recommendation, the orthotropic sandwich composite plate model developed here, should b e adopted for hydromat tests. This Whitney derived modeling approach is general and can be simplified to represent isotropic sandwich composite plates and purely isotropic plates. Thus, making the model an excellent choice for a general hydromat plate solution. 4.6. Symbols for Hydromat Modeling rm 0 z zc Ei E2 V12 V21 Qy
= = = = = = = = =
non-dimensional specially orthotropic approximation factor lamina orientation angle through thickness coordinate designates core coordinates elastic modulus (fiber direction) elastic modulus (2-direction) Poisson's ratio (12-direction) Poisson's ratio (21-direction) lamina stiffness (lamina coordinates)
Qr
=
lamina stiffness (global coordinates)
Ay By Dy A44 A55 N M Qx Qy en S22 Y12
= = = = = = = = = = = =
in-plane stiffness matrix bending extension coupling matrix bending stiffness matrix transverse core shear stiffness (yz direction) transverse core shear stiffness (xz direction) force resultants (x,y and xy components) moment resultants (x,y and xy components) transverse shear force resultants (xz components) transverse shear force resultants (yz components) in-plane strain for the 1 direction in-plane strain for the 2 direction in-plane strain for the 12 direction
62
in-plane stress in the 1 direction in-plane stress in the 2 direction in-plane stress in the 12 direction curvature in the x direction curvature in the y direction curvature in the xy direction in-plane strain for the x direction in-plane strain for the y direction in-plane strain for the xy direction transverse shear strain in the global yz direction transverse shear strain in the global xz direction in-plane stress in the x direction in-plane stress in the y direction transverse stress in the global zz direction in-plane stress in the xy direction transverse shear stress in the global xz direction transverse shear stress in the global yz direction in plane displacement in the x-direction in plane displacement in the y-direction out of plane displacement in the z-direction rotations in the x-direction rotations in the y-direction load function coefficient step load function bladder pressure reaction force from load cell iteration index associated with the x-direction (Navier's solution) iteration index associated with the y-direction (Navier's solution) panel support span transverse core shear modulus in the xdirection transverse core shear modulus in the y-direction core thickness core shear strain in the global xz direction core shear strain in the global yz direction in-plane shear stress in the global xy direction core shear stress in the global xz direction core shear stress in the global yz direction
63
Czz
transverse stress in the global zz direction
4.7 References Cited Ahtonen, P. H. and D. Sikarskie (1998). "An Analytical and Experimental Comparision of Orthotropic Sandwich Panels using the Hydromat Test System." Composites Part B29B: 705-714. ANSYS, I. (2003). ANSYS. Canonsburg: Comercial FEM Package. ASTM (2001). D6416 Standard Test Method for Two-Dimensional Flexural Properties of Simply Supported Sandwich Composite Plates Subjected to a Distributed Load. West Conshohocken, PA, American Society of Testing and Materials. Barbero, E. J. (1998). Introduction to Composite Material Design. Ann Arbor, Edwards Brothers. Bertelsen, W. D. (1992). Hydromat system. An experimental technique for the static and fatigue testing of sandwich panels. 2nd International Conference on Sandwich Constructions, Gainesville, Florida, EM AS. Bertelsen, W. D. (1994). Physical Testing of Marine Composites Using the Hydromat Test System. The Society of Naval Architects and Marine Engineers, Ann Arbor, MI. Bertelsen, W. D. (2000). Development and Certification of the ASTM D 6414-99 Plate Flexure Test and Its Implications for the Future of Composite Panel Design and Construction, Bay City, MI, Gougeon Brothers. Hyer, M. W. (1997). Stress Analysis of Fiber-Reinforced Composite Materials. Boston, McGraw-Hill. Rau, C. S., W. D. Bertelsen, et al. (1994). On the Development of a Two-Dimensional Test Fixture for Composite Panels. Fifth International Conference on Marine Applications of Composite Materials, Melbourne FL., Composite Education Association Inc. Reddy, J. N. (1984). Energy and Variational Methods in Applied Mechanics. New York, Wiley-Interscience Publications. Whitney, J. M. (1987). Structural Analysis of Laminated Anisotropic Plates. Lancaster, PA, Technomic Publishing Company, Inc. Zhao, H. (2002). Stress Analysis of Tapered Sandwich Panels with Isotropic or Laminated Composite Facings. Mechanical Engineering. Orono, University of Maine.
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5. INVERSE METHOD TO OBTAIN PANEL STIFFNESS PARAMETERS BASED ON NUMERICAL OPTIMIZATION TECHNIQUES 5.1. Abstract A combined experimental and numerical optimization approach for determining bending and shear stiffness parameters of a sandwich composite panel evaluated with the hydromat test was developed. Currently a solution exists for determining the bending and shear stiffness parameters of an isotropic sandwich panel when tested with the hydromat test system. This chapter presents and verifies a numerical optimization method for computing the bending and shear stiffness parameters of an orthotropic sandwich composite panel using the hydromat test system. First, the solution convergence was verified using simulated full-field data with initial stiffness parameter seed values containing significant error. Second, the displacement and strain data fields were perturbed using additive random noise to determine the maximum noise level corresponding to permissible magnitude of error in convergence. With this verification approach, the robustness of the solution technique was quantified. 5.2. Introduction Sandwich composite panels are constructed by bonding thin high stiffness face sheets to a low stiffness but relatively thick core material. The theory behind sandwich construction is that the relatively stiff face sheets carry the majority of the bending moments and axial loads applied to the structure. The core material stabilizes the face sheets and carries all transverse shear forces. As the core material becomes thicker, the sandwich plate becomes stiffer in shear and bending. Plate shear stiffness parameters are directly proportional to the core thickness while bending stiffness parameters are
65
approximately proportional to the thickness to the second power (e.i., I-beam or A.d2 effect). Typical sandwich construction structures exhibit a very high transverse stiffness when compared to materials with similar weight. This high stiffness to weight ratio makes sandwich construction a very attractive design option in weight critical structures. Standardized test methods have been developed to characterize sandwich composite panels and their constituents. Structural ASTM tests for sandwich construction include the Hydromat Test System (HTS), and three and four-point beam bending (ASTM 2000b). Additionally, ASTM test methods for determining properties of the sandwich composite constituents, such as tensile tests for face sheet materials and core shear tests, are available (ASTM 1998; ASTM 2000b; ASTM 2000c; ASTM 2002). This chapter presents a method to determine all the relevant elastic material properties for a sandwich composite panel using the hydromat test system. The hydromat test system was developed as a structural test for simply supported sandwich composite panels under uniform pressure (Bertelsen 1992; Bertelsen 1994; Bertelsen 2000). This test configuration matches the traditional Navier's closed form solution allowing experimental data to be correlated with analytical solutions. Previous work has shown that the HTS accurately creates the simply supported boundary conditions and uniform loading case (Rau, Bertelsen et al. 1994). ASTM D6416 states that a calibrated hydromat test system should result in plate displacements within 1% and strains within 2% of predicted values when testing an isotropic steel calibration plate (ASTM 2001). The excellent agreement between the HTS and analytical sandwich plate solutions provides an opportunity for developing an inverse solution method. Currently the ASTM standard D6416 incorporates an inverse solution method for determining the
66
bending and shear stiffness parameters of an isotropic sandwich composite plate (ASTM 2001). The method presented in this chapter expands on the HTS capability by using definitions from orthotropic sandwich composite plate theory combined with full-field data to pose and solve an inverse problem. This method allows for the reliable computation of the bending and shear stiffness parameters of sandwich composite panels.
5.3. Inverse Solution Method The methodology used in the inverse solution technique was to correlate data obtained through hydromat tests to the analytical solution. Parameters in the analytical solution were varied using a constrained non-linear optimization method until the model was the best fit to the hydromat test results. The objective function for the optimization process computed the residuals between the experimental data and the analytical solution in a least squares sense, as is shown in (5.1).
$ =^K,-Mj x
(5-1)
y
Each component of the objective function was normalized by the maximum absolute value for each experimentally measured quantity as shown in equation (5.2). Normalizing the residuals creates and equal weight for each component of the objective function. It is important to note that the objective function is a matrix with each column of the matrix containing a residual relating to an experimental quantity. Each row of the objective function represents the measured experimental quantities at the point on the hydromat test coupon determined by the coordinates (x,y). Thus, when the objective
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function is minimized, the model is the best possible match to the experimental data set as a whole.
Normalized Data =
Data Max(ABS(Data))
(5.2)
The optimization process varies the six stiffness parameters, shown in (5.3), that define the sandwich composite plate model. The optimization routine continues until the magnitude of the residuals reaches a specific tolerance. To ensure a feasible solution, constraints were applied to the stiffness parameters to guarantee that they are positive quantities. The flow chart of the optimization process is shown in Figure 5.1. Dn,D,2,D 2 2,D 6 6 ,A44,A 5 5
(5.3)
By inspection of the problem, it was found that there were six unknown stiffness parameters and only four typically measured data sets as shown in (5.4). ForEach(x,y)
M>,ex,sy,yxy
It was thought that a system with four known quantities and six unknown stiffness parameters would be indeterminate. To help ensure a unique solution, two additional experimental quantities were added by computing the gradient of the displacement field. The data sets used for the optimization routine are shown in table Table 5.1. By equating the number of experimental quantities to the number of unknown stiffness parameters, the optimization problem should become more determinate (i.e., 6 unknowns with 6 equations). The six stiffness parameters used to optimize the solution are shown in (5.3). To solve the system a Matlab routine was implemented using a non linear least squares optimization function (Isqnonlin). This function optimizes the solution using either a
68
Levenberg Marquardt or a Gauss-Newton algorithm (MathWorks 2003). The Levenberg Marquardt algorithm was selected for the optimization process. The Matlab code for the optimization process is available in the appendix. Table 5.1 Optimization Routine Input Data from Full Field Data Input Data for Each (x,y)
Normalized Input Data for Each (x,y)
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Modeling Input: Load, Pressure, Iter, a, z, hc
Initial Guess: D„, D,2, D22, D66 A44,
AS5
Geometry: Panel Thickness Core Thickness
Sandwich Composite Panel Modeling (FSDT)
Test Data Input: x, y, w, dw/dy, dw/dx, sx, ey, yxy
Compute Residuals from the Objective Function
Yes
D,j, Dl2, D22, D66 A44, AS5
Continue Optimization
Figure 5.1 Flow Chart for HTS Inverse Solution Method 5.4. Verification The inverse solution method presented in the prior section was verified by simulating input data and observing the convergence of the numerical optimization algorithm. The simulated input data was created using the analytical plate closed form solution. The input parameters used to simulate the hydromat test are shown in Table 5.2.
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Table 5.2 Physical Constants and Elastic Properties for Hydromat Test Model Physical Constants A 0
Values 500 mm 100 mm
H
Elastic Properties En
Values
E22
140 GPa lOGPa
25 mm
G12
4 GPa
t,
1 mm
N, 2
0.25
P
222,222 Pa
G,3
50 GPa
F
20,000 N
G23
100 GPa
As expected, the numerical optimization algorithm converged when the data was clean and totally unperturbed illustrating that the solution is unique. The convergence of each stiffness parameter for the "clean data" is presented in Table 5.3. It is important to note that the seed values used for the numerical method were 40% lower than the actual solution and the convergence was still excellent. Table 5.3 Optimization Results (No Noise) Stiffness Parameter
Seed Value
Actual Value
Computed Value
Difference
D„(Nm)
28560
47600
47680
0.17%
D22 (Nm)
2040
3400
3402
0.06%
D12 (Nm)
509.4
849
845
0.44%
D66 (Nm)
810
1350
1339
0.79%
G13 (MPa)
30
50
50.0
0.07%
G23 (MPa)
60
100
99.8
0.19%
%
To illustrate the numerical method's robustness, the simulated input data was perturbed using various levels of random additive noise. The "noisy" simulated data was
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created by adding positive and negative random numbers to the data. The maximum value of the random noise was defined as a percentage of the maximum experimental quantity in the simulated data set. The convergence of the solution for the "noisy" data with two levels of random noise is shown in Table 5.4 and Table 5.5. Table 5.4 Optimization Results (25% Noise) Stiffness Property
Seed Value
Actual Value
Computed Value
Difference
Dn (Nm)
28560
47600
47671
0.15%
D22 (Nm)
2040
3400
3417
0.49%
D,2 (Nm)
509.4
849
847
0.20%
D66 (Nm)
810
1350
1346
0.32%
Gn (MPa)
30
50
50.0
0.06%
G23 (MPa)
60
100
103
3.02%
%
Table 5.5 Optimization Results (50% Noise) Stiffness Property
Seed Value
Actual Value
Computed Value
Difference
Dn (Nm)
28560
47600
48419
1.72%
D22 (Nm)
2040
3400
3423
0.68%
D12 (Nm)
509.4
849
726
14.5%
D66 (Nm)
810
1350
1272
5.81%
G13 (MPa)
30
50
50.4
0.72%
G23(MPa)
60
100
99.2
0.84%
%
It was found that the numerical method had difficulties converging for the simulated data with random noise greater than 50% as shown in Table 5.5. A plot of a 50% random noise added to the transverse displacement is depicted in Figure 5.1.
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Transverse Displacement with 50% Noise
x-coordinate (meters)
Figure 5.2 Simulated Transverse Displacement Data with 50% Noise A plot of the solution convergence is illustrated in Figure 5.3. This plot was created with random input noise which explains its jagged shape. The large spikes in the error magnitude represent a divergence from the expected solution. The error magnitude was defined as the average of the percent difference between the expected results and the actual results for each stiffness parameter. As stated above, the program used to generate this plot is listed in the Appendix. Typically, the majority of the solution error was found in the Di2 and D66 stiffness terms; when these two terms deviated the solution errors in all other stiffness terms increased. It is unclear if the instability in the D12 and D65 terms is caused by local minima or a true instability in the optimization method. An error tolerance was set at 5%. The optimization routine successfully converged to the 5% error threshold for levels of random noise up to 50%.
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^
8
I
20
40
60
80
100
120
Random Noise Level (Maximum Magnitude in %)
Figure 5.3 Optimization Solution Error versus Random Noise Level 5.5. Conclusions and Recommendations A combined experimental and numerical optimization approach for determining bending and shear stiffness parameters of a sandwich composite panel evaluated with the hydromat test system was developed in Chapter 5. The solution was verified using simulated full-field data, which was perturbed using additive random noise. Based on this verification approach, the numerical optimization method was proven to be robust. Additional relationships between stiffness parameters from material coupon tests should be investigated to improve the stability of the solution (e.g, D12/D22 and D66). Additionally, it was found that the solution converged for initial seed values containing significant error from the actual stiffness parameters. An extension of this research
74
would be to calculate micromechanics properties of each lamina in the face sheets based on Dn, Di2, D22, D66. This method could improve the reliability of the calculations since fiber properties are typically well known.
5.6 References Cited ASTM (1998). D5379 Standard Test Method for Shear Properties of Composite Materials by the V-Notched Beam Method. West Conshohocken, PA, American Society of Testing and Materials. ASTM (2000b). C393 Standard Test Method for Flexural Properties of Sandwich Constructions. West Conshohocken, PA, American Society of Testing and Materials. ASTM (2000c). D3039 Standard Test Method for Tensile Properties of Polymer Matrix Composite Materials. West Conshohocken, PA, American Society of Testing and Materials. ASTM (2001). D6416 Standard Test Method for Two-Dimensional Flexural Properties of Simply Supported Sandwich Composite Plates Subjected to a Distributed Load. West Conshohocken, PA, American Society of Testing and Materials. ASTM (2002). D638 Standard Test Method for Tensile Properties of Plastics. West Conshohocken, PA, American Society of Testing and Materials. Bertelsen, W. D. (1992). Hydromat system. An experimental technique for the static and fatigue testing of sandwich panels. 2nd International Conference on Sandwich Constructions, Gainesville, Florida, EM AS. Bertelsen, W. D. (1994). Physical Testing of Marine Composites Using the Hydromat Test System. The Society of Naval Architects and Marine Engineers, Ann Arbor, MI. Bertelsen, W. D. (2000). Development and Certification of the ASTM D 6414-99 Plate Flexure Test and Its Implications for the Future of Composite Panel Design and Construction. Bay City, MI, Gougeon Brothers. Math Works (2003). Matlab Optimization Toolbox. Natick. Rau, C. S., W. D. Bertelsen, et al. (1994). On the Development of a Two-Dimensional Test Fixture for Composite Panels. Fifth International Conference on Marine Applications of Composite Materials, Melbourne FL., Composite Education Association Inc.
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6. ACCURACY AND PRECISION OF 3-D DIGITAL IMAGE CORRELATION SYSTEM FOR MATERIAL TENSILE TESTS 6.1. Abstract A 3-D digital image correlation (IC) system capable of measuring 3-D displacements and in plane strain fields was evaluated. Unlike many traditional displacement and strain measurement techniques the precision and accuracy of the digital image correlation equipment can hardly be expressed in terms of absolute units because the measurements are performed on scaled digital images of the scene rather than directly on the specimens being tested. Furthermore, the accuracy and precision of the digital image analysis systems is a resultant of many external factors, which to various degrees may be controlled by the operator. Therefore, the accuracy and precision of the measurements should be evaluated individually for each test configuration and for a selected set of parameters used for image processing. In this study, the accuracy and precision of the 3-D digital image correlation system was compared with measurements obtained from resistive strain gages and an extensometer. To quantify the accuracy and precision of the IC system a set of tensile tests were performed on 7075 T651, which is a stress relieved aluminum alloy. The aluminum alloy 7075 T651 was selected because of its consistent material properties. Three sets of coupons, each consisting of eight tensile specimens, were considered. The first set of coupons was instrumented with an extensometer; the second and third coupon sets were instrumented with resistive strain gages. All tests were monitored using the digital image correlation system allowing comparison to the conventional instrumentation.
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Material properties for the test coupons were determined using data from the IC system, resistive strain gages and the extensometer. Variations in the material properties defined the precision of the systems. The accuracy of the IC system was determined by comparing to the extensometer, which was calibrated and traceable to the National Institute of Standards and Technology (NIST). Additionally, the accuracy of all systems was determined by comparing the computed material properties to published values for the aluminum alloy selected. Excellent agreement was achieved between the resistive strain gages, extensometer, and the 3-D digital image correlation system. It was found that the digital image correlation system had one of the lowest coefficients of variation among coupons and a very high accuracy when compared to calibrated instrumentation and published material properties for 7075 T651. 6.2. Literature Review Three-dimensional digital image correlation is a powerful technique for measuring full-field, non-contact strains and displacements. The research work presented in this chapter examines the accuracy and precision of a commercial 3-D digital image correlation system, ARAMIS. The IC method was developed and used to measure deformation and strains of materials under various loading regimes with sub-pixel accuracy. The method was developed in the 1980's (Ranson, Sutton et al. 1987; Bruck, McNeill et al. 1989; Vendroux and Knauss 1998). Image correlation has been already successfully applied to determine strains for a variety of materials including metals, solid wood, individual wood fibers and paper (Choi, Thorpe et al. 1991; Mott, Shaler et al. 1996; Muszynski, R. Lagana et al. 2002; Muszynski, Lagana et al. 2003), as well as resin
77
films (Muszynski, Wang et al. 2002), fiber reinforced polymer (FRP) composites (Muszynski, Lopez-Anido et al. 2000) and concrete (Choi and Shah 1997). The IC method allows determination of displacements of selected points of the mesh on the surface of the deformed specimen by comparing successive images acquired during a test and cross correlating the gray intensity patterns of the direct neighborhood of the points (the reference areas or facets). All in-plane strain components, axial and shear, which are required to calculate the deformation of the specimen's surfaces may then be obtained from analysis based on a triangular or rectangular network of points (Muszynski, Lopez-Anido et al. 2000). A detailed description of the method may be found in Bruck, McNeill et al. 1989; Choi, Thorpe et al. 1991; Choi and Shah 1997 and in Ranson, Sutton et al. 1987. In addition, out of plane displacements may be accurately measured if two cameras are arranged at an angle in front of the same specimen area so that the displacements captured at the same instant of time by two cameras are correlated to provide stereoscopic information. Such capability is provided by the ARAMIS system by GOM, MbH, used in this study and has been successfully applied by other authors to a wide range of experimental problems. Detailed description of the 3-D measurement principles, system specifications, calibration procedures, and sample applications may be found in Bergman and Ritter; Schmidt, Tyson et al. 2002; Schmidt, Tyson et al. 2002; Schmidt, Tyson et al. 2002. 6.3. Introduction to 3-D Digital Image Correlation As described above the basic principles of image correlation are to track the movements of reference pixel neighborhoods (or facets) through a series of images. The
78
dimensions of a facet are defined by the number of pixels along its edge. Using principles of photogrammetry and grayscale cross correlation each facet in an image series can be physically located in the 3-D space. The displacements are ascribed to the facet's center points. The calculation of surface strains and out of plane deformations is based on the analysis of displacements of a regular square mesh of facets distributed over a specimen surface. The characteristic mesh parameter is called "face step" and is the distance between adjacent facet center points. Facets can overlap if the facet step, is less than the facet size. With the facets located in each image set, a plane strain field can be determined using the deformation gradient and elasticity relations. The deformation gradient is computed based on relative facet motions when compared to a reference image using Gaussian least squares adjustment to square matrix of facets. By using the polar decomposition theorem, from the theory of elasticity, the in-plane strain components can be computed from the deformation gradient. The resulting accuracy and resolution of the digital image analysis systems is affected by many factors. Some, like image acquisition conditions (e.g. lighting conditions, quality and resolution of the optical system, sensors, camera-lens geometry, surface properties of the object, position of the cameras relative to the object) or robustness of the image correlation algorithm, may be harder to control. Parameters of the analysis however, may be easily adjusted to the particular experimental setup: facet size, facet step and calculation base or the size of the strain computation matrix. It is important to note that the input parameters do not significantly affect the accuracy of the image correlation.
79
Increasing facet size may improve the precision of point recognition (or determination of an individual facet displacement) without affecting the sensitivity to local strain variations. This is done at the expense of calculation time (See Figure 6.1). single facet (or pixel neighborhood)
Facet size: 3 x 3
Facet size: 7 x 7
Figure 6.1 Increase in Facet Size The precision and local accuracy of the calculated strains may be improved by increasing the facet step. This is done at the expense of sensitivity of local strain variations but with a reduction in the calculation time (Figure 6.2). facet step is the distance between adjacent facets in the mesh
+ + + + + +
i \
1 f
+
+
+
+
' sAs*
s,
>«-
+
s,
Figure 6.2 Increase in Facet Step The precision and local accuracy may be improved by increasing the calculation base. By default, the strain values for each point of the mesh are calculated from the relative change in position of its 8 neighbors (calculation base = 3). This parameter may theoretically be increased to 5 or 7 (24 and 48 neighbors respectively) or as much as it is deemed practical. This is done on expense of the calculation time and the sensitivity to local strain variation (Figure 6.3). The improvement in accuracy stated above is only local and does not seem to improve the global accuracy.
80
• • •
• • •
• • •
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•
a
•
•
» •
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Figure 6.3 Increase in the Calculation Base (Matrix Size) Typically, a larger facet mesh and mesh spacing creates more precise strain results while sacrificing the ability to resolve small strain concentrations. Changing the facet size and facet step only improves the precision of the IC and has no effect on the accuracy of the system. Conversely, a smaller strain computation matrix (mesh dimension) and facet step (spacing) can capture small strain peaks and rapidly changing strain gradients. The effect on precision when increasing the facet size and facet step is illustrated in Figure 6.4 and Figure 6.5 respectively. 0.03 -
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40
60
70
80
Facet Size (Pixels)
Figure 6.4 Increase in the Precision versus Facet Size, with a Facet Step of 13 and a 3x3 Calculation Base
81
50 Facet Stop (Pixels)
Figure 6.5 Increase in the Precision versus Facet Step, with a Facet Size of 15 and a 3x3 Calculation Base These plots were generated by performing digital image correlation on a rigid planar surface with no deformations. The measured precision was defined as the standard deviation of the computed strain fields and the accuracy was measured as the average strain value computed over the entire surface. The strain computation matrix, which is used to compute the deformation gradient, also affects the precision of the IC. By increasing the size of the strain computation matrix, the precision of the measurement improves but at a sacrifice of resolution. The effect of the size of the strain computation matrix on precision is illustrated in Figure 6.6. It is important to note that the preliminary results showed that the input parameters of the IC system do not affect the accuracy of the strain measurements; although, when materials exhibit large strain gradients, errors may become more apparent.
82
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139
BIOGRAPHY OF THE AUTHOR
Paul Melrose was born in Augusta, Maine on July 5th, 1979. He was raised in Vassalboro, Maine and graduated from Waterville High School in 1998. He attended the University of Maine and graduated with a Bachelor of Science in Mechanical Engineering in 2002. Paul will be joining the Naval Undersea Warfare Center in Newport Rhode Island and will work as a structural design engineer. He is a candidate for the Master of Science degree in Mechanical Engineering from the University of Maine in August, 2004.
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