Application of Dynamic Fracture Mechanics Concepts to Composites: Final Report 2000 by J.J. Mason Department of Aerospace and Mechanical Engineering University of Notre Dame Notre Dame, Indiana 46556 January, 2000 DISTRIBUTION STATEMENT A Approved for Public Release Distribution Unlimited 20000209 176 Structural/Solid Mechanics Laboratory Report No. 00/1 JHIC QUALITYIMSPBCXBD 1 Application of Dynamic Fracture Mechanics Concepts to Composites: FINAL REPORT 2000 J.J. Mason 1 Department of Aerospace and Mechanical Engineering University of Notre Dame Notre Dame, IN 46556 June, 1999 1Associate Professor, Principal Investigator Abstract This report summarizes the work completed in the months of June 1996-December 1999 for the principal investigator's Young Investigator Program grant. As such the report represents a final summary of work and is organized as follows. First, in a brief summary chapter, a review of the accomplishments of the work supported from June 1996 to December 1999 is given providing an overview of the successes during that period. Next, two chapters giving details of new work during the six months since the last annual report are presented. In these results an experimentally useful solution for the stress intensity factor history at the corner of an impacting punch is derived for isotropic and orthotropic materials. Then that solution is compared to an experiment using interferometry to measure the stress intensity factor history at the corner of an impacting punch in both isotropic and anisotropic materials. Good agreement between the model and the experiments is seen. Consequently, it is concluded that the solutions derived throughout the duration of this work are accurate—mathematically speaking the fundamental solutions are actually exact—and describe the loading under blunt impact quite well. With such accurate descriptions of the stress in the material under impact, future work can now focus on understanding the failure in the material under these same conditions. Contents 1 Summary of Accomplishments 3 1.1 Previously Reported Work 3 1.2 Recent Work 7 1.3 Summary of Journal Publications 8 2 Elastodynamic Analysis of Finite Punch and Finite Crack Problems in Orthotropic Materials 10 2.1 Introduction 11 2.2 Semi-infinite Punch 14 2.2.1 Governing Equations 14 2.2.2 Method of Solution • • • 15 2.2.3 Wiener-Hopf technique 20 2.2.4 Semi-infinite smooth, rigid punch 24 2.3 Finite Punch • • • 26 2.3.1 Method of solution 26 2.3.2 Stress intensity factor 28 2.4 Semi-infinite Crack 31 2.4.1 Method of Solution 31 2.5 Finite Crack 32 2.5.1 Method of Solution 32 2.5.2 Stress intensity factor 34 2.6 Results and Conclusions 38 2.6.1 Punch Problem 38 2.6.2 Crack Problem 38 3 Experimental Investigation of Dynamic Punch Tests on Isotropie and Com- posite Materials 44 3.1 Introduction 45 3.2 Elastodynamic Analysis of the Finite Punch Problem 47 3.3 Coherent Gradient Sensing (CGS) 50 3.4 Preparation and Characterization of the Composite Specimens 54 3.4.1 Tension Test of Composites 54 3.4.2 Specimen Preparation 54 3.5 Application of the CGS Method to the Quasi-Static Punch Test 57 3.5.1 Quasi-static Punch Problem. Isotropie Material 58 3.5.2 Quasi-Static Punch Problem. Orthotropic Material 63 3.6 Dynamic Punch Test 66 3.6.1 Isotropie Materials 67 3.6.2 Orthotropic Materials 72 3.7 Conclusions 75 Chapter 1 Summary of Accomplishments An analytical approach to blunt impact problems on isotropic and anisotropic materials us- ing dynamic fracture mechanics was shown to be valid over the duration of the grant. This approach used the mathematical and experimental techniques usually associated with dy- namic fracture mechanics to examine impact of a blunt object on a wide class of materials. For simplicity sake, situations were limited to two-dimensional loading and geometries. Al- though this offers some limitations on the results, it also allows both the derivation of exact solutions describing events occurring on the interior of impacted materials and the experi- mental observation of those events, neither of which is readily achieved for three-dimensional problems. In addition, the two-dimensional results offer useful insight into three-dimensional impact events and provide valuable intuition about such events. 1.1 Previously Reported Work First, through studies of impact on metals, it was shown in this work that fracture toughness can be used to characterize failure of metals by shear localization under impact conditions. Simple solutions for the stress intensity factor in the dynamic punch test were derived using known fundamental solutions from dynamic fracture mechanics. The results of these models were published as K.M. Roessig and J.J. Mason, "Dynamic Stress Intensity Factors in a Two Di- mensional Punch Test," Engineering Fracture Mechanics, 60, No. 4, pp 421-435, 1998 The conclusion from the models was that the stress intensity history around the corner of an impacting punch was more severe than that around a notch tip in a Kalthoff test. This result was significant in light of the fact that much attention has been directed toward the Kalthoff test as a measure of resistance to shear localization during impact. As it was shown by the model, actual impact can be worse than the Kalthoff test. The solution for the stress intensity factor history around the corner of a punch impacting on an isotropic material was then used to test many types of metals. It was seen that shear band initiation at the corner of a punch was quite similar to that seen at a notch tip and that initiation occurred at the same value of stress intensity factor in both cases. These results were quite surprising but none-the-less lead one to believe that a fracture mechanics approach to modeling shear localization may be possible. The results were published as K.M. Roessig and J.J. Mason, "Adiabatic Shear Localization in the Impact of Edge Notched Specimens," Experimental Mechanics, 38, No. 3, pp. 196-203, 1998 After examining isotropic materials, attention was turned to anisotropic materials and solutions for stress intensity factors in two dimensional orthotropic materials were derived. Initially, solutions using numerical methods were found. This approach involved reducing the governing equations (in this case the conservation of momentum in terms of displacement) to a Fredholm integral equation for the Laplace transform of the stress intensity factor his- tory. Although numerical inversion of the Laplace transform, found by solving the Fredholm equation numerically, can be tricky, it was successfully performed for the case of normal and shear point loads applied to a finite crack. These results were published separately; the results for normal loads appeared as C. Rubio-Gonzalez and J.J. Mason, "Green's Functions for the Stress Intensity Factor Evolution in Finite Cracks in Orthotropic Materials," to appear Int. J. Fracture, 1999 and the results for shear point loads appeared as C. Rubio-Gonzalez and J.J. Mason, "Response of Finite Cracks in Orthotropic Materials due to Concentrated Impact Shear Loads," J. Applied Mechanics, 66, no. 2, pp. 485-491,1999 Once the problems for finite cracks had been solved it was discovered that instead of reducing the governing equations to a Fredholm integral equation, one could reduce them to a Weiner-Hopf equation. This is quite an advance in the analysis techniques for dynamic fracture. When studying isotropic materials, it is common to reduce important, fundamental problems to a Weiner-Hopf equation by using Helmholtz displacement potentials. Using a similar technique on anisotropic materials was not possible because proper Helmholtz po- tentials do not exist for anisotropic materials; they only exist for isotropic materials. Here, however, we were able to reduce the governing equations for displacements to a Weiner-Hopf equation directly, without the use of displacement potentials. Hence, a whole new class of fundamental problems in dynamic fracture mechanics could be solved. We published a sum- mary of our method, as it is applied to uniformly loaded, semi-infinite cracks in orthotropic materials, in the prestigious Journal of Mechanics and Physics of Solidsas C. Rubio-Gonzalez and J.J. Mason, "Closed Form Solutions for the Dynamic Stress Intensity Factor at the Tip of Uniformly Loaded Semi-infinite Cracks in Orthotropic Materials," to appear J. Mechanics and Physics of Solids, 1998 Then we looked at point loads applied to semi-infinite cracks in orthotropic materials. These closed form solutions serve as Green's functions for solutions to a wide class of problems and differ from our earlier numerical solutions in that they are closed form and apply to semi- infinite cracks rather than finite cracks. The new method of reducing the problem to a Weiner-Hopf equations was used to find solutions for normal point loads, published as as C. Rubio-Gonzalez and J.J. Mason, "Dynamic Stress Intensity Factor Due to Concentrated Normal Loads on Semi-infinite Cracks in Orthotropic Materials," to appear Journal of Composite Materials, 1999 and shear point loads, published as C. Wang, C. Rubio-Gonzalez and J.J. Mason, "Dynamic Stress Intensity Factor on Semi-infinite Cracks in Orthotropic Materials Due to Concentrated Shear Impact Loads," submitted to Int. J. of Solids and Structures, 1998. These solution, in contrast to the solutions described in the preceding paragraph, are closed form and not nearly as tricky to evaluate numerically. Next, we turned our attention to propagating cracks. The same method allowed us to find two approximations which describe the variation of the stress intensity factor as a function of crack velocity, one for isotropic materials that agreed with known solutions and one for highly orthotropic materials that contributed new knowledge in the area of dynamic fracture. This work was published as C. Rubio-Gonzalez and J.J. Mason, "Dynamic Stress Intensity Factor for a Prop- agating Semi-Infinite Crack in Orthotropic Materials," to appear Int. J. of En- gineering Science, 1998. Such results are useful in characterizing material resistance to dynamic crack propagation. Having found a new mathematical technique for solving two dimensional dynamic frac- ture problems in orthotropic materials, we attempted to apply the technique to quasi-three- dimensional problems. We examined the loading of penny shaped cracks in transversely isotropic materials. Using an approach based on the approach derived for two-dimensional problems, we were able to approximate the penny shaped crack problem by a simpler problem that could be reduced to a system of two Weiner-Hopf equations. This problem was solved and the results show how a penny shaped crack might grow if it were loaded dynamically in tension or shear. The paper resulting from the analysis is published as 6 C.Y. Wang and J.J. Mason, "The Dynamic Stress Intensity Factor and Strain Energy Release Rate for a Semi-infinite Crack in Rotated Transversely Isotropie Materials due to Uniform Impact Loading," submitted to Int. J. of Fracture, 1999. This results is useful when studying the initiation of delaminations in composites under impact loading. 1.2 Recent Work Finally, during the last six months of the funded effort, experimental investigation of the application of the solutions for stress intensity factors in orthotropic materials to dynamic crack initiation in composites under impact conditions was completed. First, the fundamen- tal solutions derived for semi-infinite cracks in orthotropic materials were used to calculate the stress intensity factor in the dynamic punch test. This work is presented in Chapter 2 and has been submitted as C. Rubio-Gonzalez and J.J. Mason, "Elastodynamic Analysis of Finite Punch and Finite Crack Problems in Orthotropic Materials," submitted to Theoretical and Applied Fracture Mechanics, 1999 With that solution in hand, we performed experiments on isotropic and orthotropic materials and measured the stress intensity factor history in the dynamic punch tests. This work is presented in Chapter 3 and published as C. Rubio-Gonzalez and J.J. Mason, "Experimental Investigation of Dynamic Punch Tests on Isotropic and Composite Materials," submitted to Experimental Mechanics, 1999 The results agree quite well with the solution in Chapter 2 and thus verify solutions presented here and the method used to derive them.