River Rapids Conference Proceedings of the Society for Experimental Mechanics Series Fracture, Fatigue, Failure and Damage Evolution, Volume 7 Jay Carroll Shuman Xia Alison M. Beese Ryan B. Berke Garrett J. Pataky Proceedings of the 2017 Annual Conference on Experimental and Applied Mechanics River Publishers
Conference Proceedings of the Society for Experimental Mechanics Series Series Editor Kristin B. Zimmerman, Ph.D. Society for Experimental Mechanics, Inc., Bethel, CT, USA
River Publishers Jay Carroll • Shuman Xia • Alison M. Beese • Ryan B. Berke Garrett J. Pataky Editors Fracture, Fatigue, Failure and Damage Evolution, Volume 7 Proceedings of the 2017 Annual Conference on Experimental and Applied Mechanics
Published, sold and distributed by: River Publishers Broagervej 10 9260 Gistrup Denmark www.riverpublishers.com ISBN 978-87-7004-962-7 (eBook) Conference Proceedings of the Society for Experimental Mechanics An imprint of River Publishers © The Society for Experimental Mechanics, Inc. 2018 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, or reproduction in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Preface Fracture, Fatigue, Failure and Damage Evolution represents one of nine volumes of technical papers presented at the 2017 SEM Annual Conference and Exposition on Experimental and Applied Mechanics organized by the Society for Experimental Mechanics and held in Indianapolis, IN, June 12–15, 2017. The complete proceedings also includes volumes on Dynamic Behavior of Materials; Challenges in Mechanics of Time-Dependent Materials; Advancement of Optical Methods in Experimental Mechanics; Mechanics of Biological Systems, Materials and Other Topics in Experimental and Applied Mechanics; Micro- and Nanomechanics; Mechanics of Composite, Hybrid and Multifunctional Materials; Residual Stress, Thermomechanics and Infrared Imaging, Hybrid Techniques and Inverse Problems; and Mechanics of Additive and Advanced Manufacturing. Each collection presents early findings from experimental and computational investigations on an important area within experimental mechanics, fracture and fatigue being one of these areas. Fatigue and fracture are two of the most critical considerations in engineering design. Understanding and characterizing fatigue and fracture has remained as one of the primary focus areas of experimental mechanics for several decades. Advances in experimental techniques, such as digital image correlation, acoustic emissions, and electron microscopy, have allowed for deeper study of phenomena related to fatigue and fracture. This volume contains the results of investigations of several aspects of fatigue and fracture such as microstructural effects, the behavior of interfaces, the behavior of different and/or complex materials such as composites, and environmental and loading effects. The collection of experimental mechanics research included here represents another step toward solving the long-term challenges associated with fatigue and fracture. Albuquerque, NM, USA Jay Carroll Atlanta, GA, USA ShumanXia State College, PA, USA Alison M. Beese Logan, UT, USA Ryan B. Berke Clemson, SC, USA Garrett J. Pataky v
Contents 1 Interface Mechanical Strength and Elastic Constants Calculations via Nano Impact and Nanomechanical Raman Spectroscopy .......................................................................... 1 Devendra Verma and Vikas Tomar 2 Effect of Strain Rate and Interface Chemistry on Failure in Energetic Materials ............................... 7 Chandra Prakash, I. Emre Gunduz, and Vikas Tomar 3 Characterization of Crack Tip Plasticity in IN-617 Using Indentation and Nano-Mechanical Raman Spectroscopy ............................................................................................................. 13 Yang Zhang and Vikas Tomar 4 The Two-Way Relationship Between Residual Stress and Fatigue/Fracture ..................................... 19 Michael B. Prime 5 Designing Brittle Fracture Specimens to Investigate Environmentally Assisted Crack Growth ............... 25 Sunday Aduloju, Wenjia Gu, Timothy Truster, John Emery, Dave Reedy, and Scott J. Grutzik 6 Flexible Energy Harvesting/Storage Structures for Flapping Wing Air Vehicles ................................ 35 Alex Holness, Hugh A. Bruck, and Satyandra K. Gupta 7 The Influence of Formulation Variation and Thermal Boundary Conditions on the Near-Resonant Thermomechanics of Mock Explosives................................................................................ 47 Allison R. Range, Nicole R. McMindes, Jaylon B. Tucker, and Jeffrey F. Rhoads 8 Detecting Fatigue Crack Closure and Crack Growth Delays After an Overload Using DIC Measurements . 57 G.L.G. Gonzáles, J.A.O. González, J.T.P. Castro, and J.L.F. Freire 9 In-Situ Observation of Damage Evolution in Quasi-Isotropic CFRP Laminates ................................ 67 Addis Tessema, Suraj Ravindran, Abigail Wohlford, and Addis Kidane 10 Contamination-Induced Degradation/Enhancement of Interfacial Toughness and Strength in Polymer-Matrix Composite Interfaces ............................................................................. 73 Denizhan Yavas, Xu Shang, and Ashraf F. Bastawros 11 Direct and Simultaneous Extraction of Mixed-Mode Traction-Separation Relations ........................... 79 Chenglin Wu, Rui Huang, and Kenneth M. Liechti 12 Damage Evolution in 304L Stainless Steel Partial Penetration Laser Welds ..................................... 85 Sharlotte Kramer, Amanda Jones, John Emery, and Kyle Karlson 13 Cross-Axis Coupling and Phase Angle Effects Due to Multiaxial Vibration...................................... 95 Ed Habtour, Abhijit Dasgupta, and Sabrina Vantadori 14 Behavior of Steel-Concrete Composite Beams Under Fatigue Loads.............................................. 99 Ayman El-Zohairy and Hani Salim 15 Studying the Fracture of Tropical Wood Species with the Grid Method.......................................... 111 B. Odounga, R. Moutou Pitti, E. Toussaint, and M. Grédiac vii
viii Contents 16 Generalization of Integral Parameters to Fatigue Loading in Room Temperature.............................. 115 Rostand Moutou Pitti, Hassen Riahi, and Mulugeta A. Haile 17 Fracture Behavior of Unidirectional Composites Analyzed by Acoustic Emissions Technique ................ 121 C. Barile and C. Casavola
Chapter 1 Interface Mechanical Strength and Elastic Constants Calculations via Nano Impact and Nanomechanical Raman Spectroscopy Devendra Verma and Vikas Tomar Abstract Interfaces are ubiquitous in important natural and manmade materials. Research evidence has shown that interface chemistry, structure, and thickness together strongly influence material microstructure and mechanical properties. The focus of the present work is on presenting an experiment based theoretic advancement to predict thickness dependent elastic properties of materials interfaces by treating the interfaces and the area around them in a material as an elastic continuum. The experiments are based on the nanomechanical Raman spectroscopy (NMRS) developed by authors earlier with a capability to simultaneously measure stress components in orthogonal directions during an in-situ nanomechanical loading. An analytical model is developed based on boundary conditions of interface to predict thickness dependent interface elastic constants. The interface elastic constants are compared with the relations provided in literature. Keywords Nanomechanical Raman spectroscopy • In-situ interface deformation • In-situ nanomechanical measurements • Interface thickness • Interface elastic constants The first ever mention of interfaces is in the work of Gibbs [1] where he formulated thermodynamic foundations of interface excess energy. Gibbs definition of interfaces was a zero thickness mathematical entity. The focus of the present work is different from interface thermodynamics, interface chemistry, and interface structure characterization work available in literature. The interface in this study is considered to be of finite thickness with a non-zero volume. Emphasis is on deducing the influence of interfaces on mechanical deformation using a classical approach that incorporates interface multi-axial properties. The present work uses a nanomechanical Raman spectroscopy (NMRS) [2–7] based experimental framework to measure direct in-situ interface deformation properties. In the classical work by Dingreville and Qu [8–10] the interfacial mismatch stress was related to the in-plane strain and applied stress in the case of a zero thickness interface. These formulations provide a way to calculate interfacial elastic properties based on the contribution of interfacial coherent surface stress, incoherent surface stress and transverse excess strain. The role of transverse direction properties of interfaces is accounted for mathematically. In an another formation, the interface is explicitly considered as a finite thickness entity to calculate the interface elastic constants in the case of heterogeneous thin interfaces by Ustinov et al. [11] showing the interface stiffness dependence on their thickness. The present work focuses on using NMRS based direct observations of multiaxial interface deformation to develop a theoretical framework for predicting interface elastic constants as a function of interface thickness. Interfaces in composite materials are considered as a material phase confined between two separate grains or phases. In this experimental work, the interface elastic constants are measured in the case of an idealized epoxy interface between two glass plates. Single interface samples of glass and epoxy were prepared with an epoxy interface sandwiched between two glass plates. The thickness of the interface in samples was measured with a microscope to make sure that it was in the error margin of 10 ˙0.5 m. Considering different stiffness for the interface and the bulk phases, this boundary condition leads to a jump in the stresses. It has continuous strain distribution for the component in indentation direction while the stresses perpendicular to the interface must fulfill the condition of equilibrium. To solve this problem, a fictitious homogeneous isotropic elastic half space is assumed which for a given applied load through the indenter exhibits the same normal surface displacement as the material with interface. Figure 1.1 shows the schematic of solution procedure for this case. The key assumption of the analytical approach presented here is that the classical strain solution and the solution for the layered half space are similar for the applied loads. The assumptions for these components are therefore: D. Verma • V. Tomar ( ) School of Aeronautics and Astronautics, Purdue University, West Lafayette, IN, 47907, USA e-mail: vermad@purdue.edu; tomar@purdue.edu © The Society for Experimental Mechanics, Inc. 2018 J. Carroll et al. (eds.), Fracture, Fatigue, Failure and Damage Evolution, Volume 7, Conference Proceedings of the Society for Experimental Mechanics Series, DOI 10.1007/978-3-319-62831-8_1 1
2 D. Verma and V. Tomar 1. Fictitious homogeneous half space 3. Some strain and some stress component distributions show a good match. ≈ E =? n =? 4. Get the stresses for the layered half space with using material laws 2. Solution from the Literature u, ,σ σxx σzz σxz zz, xz, σxx zz, xz, σxx Fig. 1.1 Schematic illustration of the solution procedure xx;fict xx;lay; xz;fict xz;lay; xy;fict xy;lay: (1.1) The remaining strain components in interface plane direction, and must fulfill the conditions of compatibility and strain jumps cannot occur. yy;fict yy;lay; yz;fict yz;lay: (1.2) Due to limitations of the stress measuring technique, it is not possible to obtain a full stress tensor for the interface in the current indentation setup. Nevertheless, an equivalent stress can be obtained, therefore the calculated stress tensors are transferred to an equivalent stress by: v Dq 2 xx C 2 yy C 2 zz xx yy C yy zz C xx zz C3 2 xy C 2 yz C 2 xz : (1.3) Here, v is the von Mises equivalent stress. The flat punch corrections were then applied in the model. A flat ended and a spherical indenter produce different load distributions on the surface of the sample. To validate the assumptions of the model, an interface FE model and a homogeneous isotropic half space model was simulated with 10 m thick interface in the middle. Plane strain loading boundary conditions were applied and the loading was given in the displacement boundary condition. The strains were measured at the maximum displacement of 500 nm that was equivalent to the indentation depth in the actual experiments. The strains and stresses were compared for both models. The stresses showed the similar profile validating the stress assumptions given in Eq. (1.1). Raman spectroscopy is an excellent tool to measure properties such as the crystalline structure, chemical signature without a necessity of sample preparation. The Raman spectroscopy has been used for other material systems such as epoxy in recent years to measure the curing state as well as the residual stresses in the sample44. We have used the Raman spectroscopy developed in our lab by Gan and Tomar to measure the stresses in the interface at different applied loads during nanomechanical loading to compare the stress distribution [3, 4]. The mechanical load was applied using the nanoindentation platform manufactured by Micro Materials Inc., UK, [12, 13] with load range from 0.1 to 500 mN, with the accuracy of 0.01 mN. The nanomechanical loading was performed using a flat punch indenter attached to the pendulum.
1 Interface Mechanical Strength and Elastic Constants Calculations via Nano Impact and Nanomechanical Raman Spectroscopy 3 Fig. 1.2 (a) Setup of the nanomechanical Raman spectroscopy experiments. (b) Raman spectrum collected from epoxy showing the peak corresponding to C–H bond. (c) Raman shift versus stress calibration curve for epoxy Fig. 1.3 Distribution of the equivalent stress for the indentation with a flat indenter for 20, 200, and 500 mN. The height in each map is 80 m and width is 10 m The epoxy samples show Raman peaks in a wide range from 560 to 645 nm. The Raman signal at each wavelength depends on the mass of the atoms involved and the strength of the bonds between them. In the current system we measured the strongest signal around 641.1 nm as shown in Fig. 1.2b. The change in shift was obtained by subtracting the shift at the applied load for the shift at zero load. The calibration curve for shift versus load for epoxy is given in Fig. 1.2d. The Raman shift versus stress calibration curve was used to calculate the stress maps on the interfaces. The Raman maps were measured at 0, 6, 60, and 150 MPa. The measurements were performed while holding the load constant. The stress distribution across interfaces is shown in Fig. 1.3. The Raman spectroscopy only provides the average stress at the interface but stress tensors in different directions are needed to fully understand the behavior of interfaces. Even in the experiments, it is difficult to measure the lateral stresses. An analytical solution is therefore developed to calculate the lateral stresses during indentation of interfaces. In the present analyses, the indentations are quasistatic and fully elastic which gives small indentation depths compared to the indenter
4 D. Verma and V. Tomar Table 1.1 Elastic constants for glass/epoxy interface A11 (Pa-m) A22 (Pa-m) A33 (Pa-m) A13 (Pa) Thiswork 34,029 6821 2.03 1014 2.80 109 Theoretical approximations [11] 53,571 53,571 5.35 1014 3,571,428,571 radius. This condition allows for the simplifications of the loaded region. An analytical method to calculate the stresses in interfaces with vanishing thickness is presented by Ustinov et al. [11]. In this work, we replace the intermediate layer in the internal energy equation given in [11] by a layer of thickness h, which is less than the thickness of two original layers. A new two-layered system with additional interface elastic constants is obtained with its elastic energy as a function of the thickness of the intermediate layer. While the longitudinal strains stay the same as for the initial system, a new relation for the transverse strains are presented, by claiming that the surface displacements should coincide with the initial system. By substituting equation of elastic energy of the modified system expressed in terms of the strains of the initial system and solving the resulting equation system which results by equating after performing the limit transition the elastic properties of the interface with vanishing thickness can be obtained [14]. The abovementioned model was programmed in a MATLAB code to calculate the interface stresses for different scenarios, [14]. The stresses were then calculated for the quasistatic case for the same applied load as in the experiments. The values from the Nanomechanical Raman spectroscopy measurement and the analytical solution are of the same order showing the validity of the model to measure the stress components in the given case, Fig. 1.3. The stress and strains were then calculated in all direction using the analytical model to calculate the interface elastic constants. The elastic constants were then calculated from these stress-strain data using the linear fit. These formulas were further used to calculate the interface elastic constants of the epoxy interfaces analyzed in the current study. The interface elastic constants for epoxy interface measured from the indentation experiments after conversion to surface constant is listed in the same Table 1.1 with the values obtained from theoretical approximation for comparison. A new formulation based on the NMRS is presented to calculate the interface elastic constants using an analytical model. The measured and calculated elastic constants are compared with the strain energy frameworks provided in the literature. A comparison between the current and literature methods shows the dependence of the interface elastic constants on the interface thickness. The elastic constants calculated from the stress-strain data match the literature values after the thickness effect correction. References 1. Gibbs, J.W.: The Collected Works of J. Willard Gibbs, Volume I: Thermodynamics. Yale University Press, New Haven (1928) 2. Gan, M., Samvedi, V., Tomar, V.: Raman spectroscopy-based investigation of thermal conductivity of stressed silicon Microcantilevers. J. Thermophys. Heat Transf. 29(4), 845–857 (2014) 3. Gan, M., Tomar, V.: An in situ platform for the investigation of Raman shift in micro-scale silicon structures as a function of mechanical stress and temperature increase. Rev. Sci. Instrum. 85(1), 013902 (2014) 4. Gan, M., Tomar, V.: Surface stress variation as a function of applied compressive stress and temperature in microscale silicon. J. Appl. Phys. 116(7), 073502 (2014) 5. Zhang, Y., Gan, M., Tomar, V.: In-Situ Combined Sensing of Uniaxial Nanomechanical and Micromechanical Stress with Simultaneous Measurement of Surface Temperature Profiles by Raman Shift in Nanoscale and Microscale Structures. Purdue Research Foundation, West Lafayette (2016) 6. Zhang, Y., Gan, M., Tomar, V.: Small scale thermomechanics in Si with an account of surface stress measurements. In: Ralph, C., et al. (eds.) Mechanics of Composite and Multi-Functional Materials, Volume 7: Proceedings of the 2015 Annual Conference on Experimental and Applied Mechanics, pp. 247–250. Springer International Publishing, Cham (2016) 7. Zhang, Y., Gan, M., Tomar, V.: Raman thermometry based thermal conductivity measurement of bovine cortical bone as a function of compressive stress. J. Nanotechnol. Eng. Med. 5(2), 021003 (2014) 8. Dingreville, R., Hallil, A., Berbenni, S.: From coherent to incoherent mismatched interfaces: a generalized continuum formulation of surface stresses. J. Mech. Phys. Solids. 72(0), 40–60 (2014) 9. Dingreville, R., Qu, J.: Interfacial excess energy, excess stress and excess strain in elastic solids: planar interfaces. J. Mech. Phys. Solids. 56(5), 1944–1954 (2008) 10. Dingreville, R., Qu, J., Mohammed, C.: Surface free energy and its effect on the elastic behavior of nano-sized particles, wires and films. J. Mech. Phys. Solids. 53(8), 1827–1854 (2005) 11. Ustinov, K.B., Goldstein, R.V., Gorodtsov, V.A.: On the modeling of surface and interface elastic effects in case of eigenstrains. In: Altenbach, H., Morozov, F.N. (eds.) Surface Effects in Solid Mechanics: Models, Simulations and Applications, pp. 167–180. Springer Berlin Heidelberg, Berlin (2013)
1 Interface Mechanical Strength and Elastic Constants Calculations via Nano Impact and Nanomechanical Raman Spectroscopy 5 12. Verma, D., Tomar, V.: Strain rate dependent failure of metallic interfaces at nano- microscale via nanoimpact experiments. In: 20th International Conference on Composite Materials, Copenhagen (2015) 13. Verma, D., Tomar, V.: An investigation into environment dependent nanomechanical properties of shallow water shrimp (Pandalus platyceros) exoskeleton. Mater. Sci. Eng. C Mater. Biol. Appl. 44, 371–379 (2014) 14. Verma, D., et al.: A Combined Theoretic and Experimental Advancement to Measure Interface Stress-Strain Relations and Interface Elastic Constants using Nanomechanical Raman Spectroscopy. Manuscript in preparation (2017)
Chapter 2 Effect of Strain Rate and Interface Chemistry on Failure in Energetic Materials Chandra Prakash, I. Emre Gunduz, and Vikas Tomar Abstract We study the failure at interfaces between Hydroxyl-terminated polybutadiene (HTPB)-Ammonium Perchlorate (AP) based energetic material. In this work, interface mechanical strength of a set of HTPB-AP interfaces is characterized using nano-scale impact experiments at strain rates up to 100 s 1. A power law viscoplastic constitutive model was fitted to experimental stress-strain-strain rate data in order to obtain constitutive behavior of interfaces, particle, and matrix. A mechanical Raman spectroscopy is used to analyze the effect of binding agent at different temperature. A tensile fracture experiment combined with In-situ Mechanical Raman Spectroscopy was used to obtain fracture properties. Stress maps are obtained near the interface using In-situ Mechanical Raman Spectroscopy to analyze the changes in the stress distribution around interfaces for different loads till failure. Cohesive zone model parameters were obtained from the consideration of local stress during failure and the cohesive energy required for delamination of AP from HTPB matrix. Effect of binding agent on the interface strength is found to be quite significant. The cohesive zone parameters and the viscoplastic model obtained from the experiment were then used in the cohesive finite element method to simulate the dynamic crack propagation as well as the delamination. Results show that interfacial properties are affected by the rate of loading and are also dependent upon the binding agent. Keywords Energetic material • Stress/strain relationship • HTPB • AP • NRS Energetic compounds are employed in a large number of applications, such as, explosive, propellant, and pyrotechnic formulations. An example of a heterogeneous solid propellant used in rocket industry is a crystalline oxidizer (e.g., ammonium perchlorate-AP) embedded in a polymeric binder (e.g., Hydroxyl-Terminated Polybutadiene or HTPB). Aluminum (Al) particles are sometimes added to enhance the propellant performance. A typical industrial solid propellant consists of 70% AP, 10% HTPB and around 20% Al by weight, [1]. Three main failure mechanisms in the composite material are identified as particle fracture, interfacial failure and the cavitation in the binder [2]. Palmer [2] investigated a number of polymer bonded explosives (PBXs) under tensile loading and observed finding a wide range of responses. They found that the crack propagation was mostly confined to binder and that the interface debonding was the dominant failure mode. Interface strength depends on the constituent material, i.e., particle, matrix and/or binding agents [3, 4]. Several experiments [5–7] have suggested a particle size effect on the performance of energetic materials. Yeager [8] has shown the effect of interface/interphase and the microstructure on the mechanical behavior of PBXs. Interfacial structure was altered by adding a plasticizer in the composite. The plasticizer was shown to inhibit the formation of a large interface/interphase and was more likely to have film delamination than the no-plasticized composite. The difference in interfacial properties was also shown to have significant effect on the crack initiation and explosive sensitivity. Two samples were prepared for analyzing the effect of functionalization on the interface mechanical properties. One consist of ammonium perchlorate (AP) particles embedded in hydroxyl-terminated polybutadiene (HTPB). In the second sample, a surface binding agent (Tepanol) was added at a mass ratio of 0.5 to fabricate samples with higher surface adhesion, while keeping the same index ratio. C. Prakash • V. Tomar ( ) School of Aeronautics and Astronautics, Purdue University, West Lafayette, IN, 47907, USA e-mail: cprakash@purdue.edu; tomar@purdue.edu I. Emre Gunduz School of Mechanical Engineering, Purdue University, West Lafayette, IN, 47907, USA © The Society for Experimental Mechanics, Inc. 2018 J. Carroll et al. (eds.), Fracture, Fatigue, Failure and Damage Evolution, Volume 7, Conference Proceedings of the Society for Experimental Mechanics Series, DOI 10.1007/978-3-319-62831-8_2 7
8 C. Prakash et al. Table 2.1 Viscoelastic material parameters for AP, HTPB and the HTPB/AP interface ¦(MPa) n m n Sample-1 Sample-2 Sample-1 Sample-2 Sample-1 Sample-2 HTPB 25.52 13.13 19 13 25 30 AP 18 19.9 16 17 9 9 HTPB/AP interface 40.3 12.8 30 10 8 3 Energetic materials have been modeled using viscoelastic [9, 10] as well as elasto-viacoplastic [11, 12] model. For high strain rate loading viscoplastic models are used frequently [13–15]. Kalayciogly [12] et al. modeled an HTPB/AP composite propellant using Perzyna’s viscoplastic model. In this work, following Tsai and Sun, [16], we assume an effective stresseffective viscoplastic strain curve by a power law model "vp DA. / n ; (2.1) where "vp and are the equivalent strain and equivalent stress respectively. A is a power law function of effective plastic strain rate given as, AD P" 0 m : (2.2) The viscoplastic model then can be given as, P" vp D P" 0 m . / n : (2.3) The values of , mand nare found from the stress strain data obtained during nano-scale impact test, [17]. Mechanical properties were obtained using a nano-scale impact experiment. The experimental procedure involves impacting the surface of material being tested by an indenter. The nano-scale impact were performed using the high strain rate impact schedule of Micro Materials, UK [18–21]. Table 2.1 shows that the viscoplastic model parameters for the interface, the matrix (HTPB) and the particle (AP) for two samples. These material parameters then further can be used to describe the material behavior at the interface as well as in the matrix and particle phase for high strain rates. The present analysis shows nano-scale impact as a tool to capture the material deformation behavior at micrometer scales. As shown in Fig. 2.1a, the samples were loaded in tension at a loading rate of 0.1 mm/min till fracture. During the loading, a set of points on the sample near interface was scanned to get Raman shifts, [22]. These Raman shifts were obtained at different loads until the debonding occurred. The strength of the interface was evaluated from the stress map obtained using Raman spectroscopy by assuming the strength equal to the stress at the start of the delamination. The strength of the interface is the stress near the interface at the load during failure (Fig. 2.1b). Area under the load displacement curve, Fig. 2.1c, between the point where crack reaches the interface and the point where interface delaminates is the total energy required for delamination. The dynamic behavior of the composite material system is modeled using the explicit finite element method (see e.g., [23, 24]). The average time step is of the order of 1 ns which is approximately one tenth of the time it takes for the longitudinal wave to traverse the smallest bulk element in the meshes in current research. The weak form of the governing equation is given as, Z V W •FdV Z Sint T:ı dS DZ Sext T:•udS Z V @2u @t2 •udV; (2.4) To track complex crack/microcrack patterns, arbitrary crack paths, and crack branching, cohesive surfaces are specified along all finite element boundaries as an intrinsic part of the finite element model. The finite element meshes used have a uniform structure with “cross-triangle” elements of equal dimensions arranged in a quadrilateral pattern, Fig. 2.2. This type of triangulation is used since it gives the maximum flexibility for resolving crack extensions and arbitrary fracture patterns [25].
2 Effect of Strain Rate and Interface Chemistry on Failure in Energetic Materials 9 Fig. 2.1 (a) Loading configuration, (b) stress distribution around the interface for different load until failure and (c) Load-displacement curve Fig. 2.2 Finite element mesh of the energetic material model being studied We use the standard assumption for finite strain inelastic problems: the multiplicative decomposition of the deformation gradient Finto an elastic and an inelastic part Fe and Fp, i.e., F DFe Fvp (2.5) In the cohesive model used, the traction T applied on material points coinciding at and occupying position x on cohesive surface in the reference configuration is work-conjugate to surface separation . Reckoned in the reference configuration, the cohesive law is T.x/ DTΠ.x/ ; (2.6) and the work of separation under this traction at any stage of deformation is [26], Wsep DZ S0 Z 0 T.x; / :d dx: (2.7)
10 C. Prakash et al. Table 2.2 Cohesive zone parameters for constituents of HTPB/AP composite Material/interface Cohesive strength (MPa) Critical displacement (mm) Cohesive energy (N/mm) HTPB 0.8 0.5 0.2 AP 200 1 10 6 1 10 4 Interface 0.5–6 0.11 0.02–0.33 Fig. 2.3 Validation of delamination mode obtained using CFEM with experiment The CFEM simulations are carried out under plane strain assumption. The plane strain assumption is therefore a major limitation of the framework. The irreversible bilinear law is used which is a generalized version of the cohesive laws with irreversibility [27, 28]. The current law is derived from a description ®of the surface energy dissipation per unit area which is a function of separation vector through a state variable defined as that describes the effective instantaneous state of mixed-mode separations. The specific form for ®is taken as, [29]. A two dimensional numerical simulation is carried out for the AP-HTPB specimen with a single AP particle embedded in HTPB binder to simulate the failure. Figure 2.1 illustrates the geometry and boundary conditions employed in the numerical simulation. The AP particle is idealized as a circle with diameter DD1 mm. The size of the element was chosen such that it satisfies the bounds given in [30], which is equal to 25 m. Local cohesive zone parameters used in the simulation were obtained from the Raman spectroscopy experiment, [17], and are given in Table 2.2. These parameters for HTPB and HTPB/AP interface were obtained from the Raman spectroscopy as explained above. For AP the cohesive strength is assumed to be E/100, [30], where E is the Young’s modulus of AP. HTPB/AP interface cohesive strength was varied to study the effect of different interface. Figure 2.3 shows the similarity in the mode of delamination observed in both numerical and experimental observations. Strain rates of 100, 200, 800, 1000, 2000, 3000 s 1 were applied in terms of the velocity boundary condition at the top boundary. Figure 2.4a shows the effect of strain rates on the cohesive energy of the system. Cohesive energy increases with the time as the deformation increases, even when the cohesive zone parameters are constant. This is because the material model used is a rate dependent viscoplastic material. The rate at which the cohesive energy increases also increase with strain rate. Delamination starts near the fix boundary at low strain rates (800 s 1 and 1000 s 1). However, as the strain
2 Effect of Strain Rate and Interface Chemistry on Failure in Energetic Materials 11 Fig. 2.4 Cohesive energy of the system (a) for different strain rates and (b) for different cohesive strength with time rate increases the delamination occurs on the loading side (3000 s 1). Figure 2.4b shows the effect of cohesive strength on the cohesive energy. The delamination occurs when the cohesive energy reaches the critical value. As can be seen in the Fig. 2.4b, time at which cohesive energy reaches its critical value, decreases with increasing cohesive strength. This work shows the effect of interface strength and the strain rate on the debonding in HTPB/AP composite. We obtain a strain rate dependent mechanical properties of constituents of the HTPB/AP composite using a nano-scale impact experiment which allows us to obtain not only the properties of constituents but also of interface which is otherwise difficult to obtain. Cohesive parameters based on in-situ mechanical Raman spectroscopy based on change in interface chemistry by adding a binding agent into the composite propellant. We then use these experimentally obtained properties into our cohesive finite element model (CFEM) to simulate the delamination of HTPB-AP interface. The dynamic fracture behavior is simulated for the HTPB/AP sample with CFEM. The CFEM simulation accurately captures the evolution and mode of delamination. This has been confirmed by comparing it with the quasi-static tensile experiment also. Strain rate effects are considered and shown to affect the delamination in the sample. The effect of interface chemistry on delamination mode can eventually effect the possible hot-spots in the material. Further study would be needed to quantify this effect in multi-particle energetic materials. Acknowledgments This research was supported by US-AFoSR Grant FA9550-15-1-0202 (Program Manager Dr. Martin Schmidt). References 1. Gallier, S., Hiernard, F.: Microstructure of composite propellants using simulated packings and X-ray tomography. J. Propuls. Power. 24(1), 154–157 (2008) 2. Palmer, S.J.P., Field, J.E., Huntley, J.M.: Deformation, strengths and strains to failure of polymer bonded explosives. Proc. R. Soc. A: Math. Phys. Eng. Sci. 440, 399–419 (1993) 3. Stacer, R.G., Hubner, C., Husband, M.: Binder/filler interaction and the nonlinear behavior of highly-filled elastomers. Rubber Chem. Technol. 63(4), 488–502 (1990) 4. Stacer, R.G., Husband, M.: Small deformation viscoelastic response of gum and highly filled elastomers. Rheol. Acta. 29, 152–162 (1990) 5. Fleming, K.A., et al.: The influence of formulation variables on the growth of reaction in plastic bonded explosives. In: Proceedings of the 8th International Detonation Symposium, Albuquerque. Naval Surface Weapons Center (1985) 6. Kimura, E., Oyumi, Y.: Shock instability test for azide polymer propellants. J. Energ. Mater. 16(2–3), 173–185 (1998) 7. Rae, P.J., et al.: Quasi-static studies of the deformation and failure of “-HMX based polymer bonded explosives. Proc. R. Soc. A: Math. Phys. Eng. Sci. 458, 743–762 (2002) 8. Yeager, J.D.: Microstructural Characterization of Simulated Plastic Bonded Explosives, in Mechanical and Materials Engineering. Washington State University, Washington (2011)
12 C. Prakash et al. 9. Wang, Z., et al.: Tensile mechanical properties and constitutive model for HTPB propellant at low temperature and high strain rate. J. Appl. Polym. Sci. 132, 42104 (2015) 10. Renganathan, K., et al.: Tensile fracture of HTPB based propellant specimens. Mater. Sci. Technol. 18(11), 1408–1412 (2013) 11. Xu, F., Aravas, N., Sofronis, P.: Constitutive modeling of solid propellant materials with evolving microstructural damage. J. Mech. Phys. Solids. 56(5), 2050–2073 (2008) 12. Kalaycioglu, B., Dirikolu, M.H., Çelik, V.: An elasto-viscoplastic analysis of direct extrusion of a double base solid propellant. Adv. Eng. Softw. 41(9), 1110–1114 (2010) 13. Trumel, H., et al.: A constitutive model for the dynamic and high-pressure behaviour of a propellant-like material: part II: model development and applications. Int. J. Numer. Anal. Methods Geomech. 25(6), 581–603 (2001) 14. Trumel, H., et al.: A constitutive model for the dynamic and high-pressure behaviour of a propellant-like material: part I: experimental background and general structure of the model. Int. J. Numer. Anal. Methods Geomech. 25(6), 551–579 (2001) 15. Trumel, H., Fanget, A., Deragon, A.: A finite strain elastic-plastic model for the quasi-static behaviour of particulate composites. Int. J. Eng. Sci. 34(6), 677–698 (1996) 16. Tsai, J., Sun, C.T.: Constitutive model for high strain rate response of polymeric composites. Compos. Sci. Technol. 62, 1289–1297 (2002) 17. Prakash, C., et al.: Strain rate dependent failure of interfaces examined via nanoimpact experiments. In: Antoun, B. et al. (eds.) Challenges in Mechanics of Time Dependent Materials, vol. 2. Conference Proceedings of the Society for Experimental Mechanics Series. Springer, Cham, pp. 93–102 (2017) 18. Verma, D., Tomar, V.: An investigation into environment dependent nanomechanical properties of shallow water shrimp (Pandalus platyceros) exoskeleton. Mater. Sci. Eng. C Mater. Biol. Appl. 44, 371–379 (2014) 19. Verma, D., Tomar, V.: A comparison of nanoindentation creep deformation characteristics of hydrothermal vent shrimp (Rimicaris exoculata) and shallow water shrimp (Pandalus platyceros) exoskeletons. J. Mater. Res. 30(08), 1110–1120 (2015) 20. Verma, D., Tomar, V.: An investigation into mechanical strength of exoskeleton of hydrothermal vent shrimp (Rimicaris exoculata) and shallow water shrimp (Pandalus platyceros) at elevated temperatures. Mater. Sci. Eng. C Mater. Biol. Appl. 49, 243–250 (2015) 21. Verma, D., Qu, T., Tomar, V.: Scale dependence of the mechanical properties and microstructure of crustaceans thin films as biomimetic materials. JOM. 67(4), 858–866 (2015) 22. Prakash, C., et al.: An analysis of the influence of grain boundary strength on microstructure dependent fracture in polycrystalline tungsten. Int. J. Fract. 199(1), 1–20 (2016) 23. Fish, J., et al.: AL 6061-T6-Elastomer impact simulation. Technical Report. Rensselaer Polytechnic Institute (2005) 24. Hui, T., Oskay, C.: Computational modeling of polyurea-coated composites subjected to blast loads. J. Compos. Mater. 46(18), 2167–2178 (2012) 25. Tomar, V.: Insights into the effects of tensile and compressive loadings on microstructure dependent fracture of trabecular bone. Eng. Fract. Mech. 76(7), 884–897 (2009) 26. Ortiz, M., Pandolfi, A.: Finite-deformation irreversible cohesive elements for three-dimensional crack-propagation analysis. Int. J. Numer. Methods Eng. 44(9), 1267–1282 (1999) 27. Tvergaard, V.: Cohesive zone representations of failure between elastic or rigid solids and ductile solids. Eng. Fract. Mech. 70(14), 1859–1868 (2003) 28. Camacho, G.T., Ortiz, M.: Computaional modeling of impact damage in brittle materials. Int. J. Solids Struct. 33(20–22), 2899–2938 (1996) 29. Zhai, J., Tomar, V., Zhou, M.: Micromechanical simulation of dynamic fracture using the cohesive finite element method. J. Eng. Mater. Technol. 126(2), 179 (2004) 30. Tomar, V., Zhai, J., Zhou, M.: Bounds for element size in a variable stiffness cohesive finite element model. Int. J. Numer. Methods Eng. 61(11), 1894–1920 (2004)
Chapter 3 Characterization of Crack Tip Plasticity in IN-617 Using Indentation and Nano-Mechanical Raman Spectroscopy Yang Zhang and Vikas Tomar Abstract This research focuses on work with emphasis on direct measurements of stresses during mesoscale microstructural deformation of nickel based super alloys during 3-point bending tests at elevated temperatures. A novel nano-mechanical Raman spectroscopy measurement platform was designed for temperature, stress, and chemistry mapping at micro to nanoscale for different temperature and loading conditions. During the 3-point bending test, notch tip plastic stresses as a function of microstructure, load, and temperature, with micron scale resolution were measured. The temperature field distribution was correlated to stress distribution and residual microstructure stresses around the area of the notch tip. Grain boundaries are the stress concentrated area but with lower temperature due to the contact thermal resistant. Grain boundary slide or grain rotation can result in stress concentration but enhance the ability of heat conduction and result in lower temperature. The mechanical properties which include the elastic modulus, hardness and stress-strain relation at the plastic zone around the notch tip were also measured. Instead of considering actual grain structures with different material properties, a new FE method was adopted to predict stress distribution applying the material properties which were obtained from indentation experiments around the same notch area as scanning. Predictions from theory and simulations matched closely in stress concentration area with experimental measurements. However, away from notch area a slight deviation due to microstructural effects was observed. Keywords Nano-mechanical Raman • Raman spectroscopy • Stress distribution • Temperature field • Nanoindentation There are several existing experimental methods to measure stress distribution at small scale such as X-ray diffraction (XRD) [1] and cross-sectional transmission electron microscopy (XTEM) [2]. Raman spectroscopy method to measure the stress was first introduced by Anastassakis et al. in 1970 to measure the mechanical stress inside silicon [3]. From then on, this method has been used and developed extensively to investigate stress distribution in silicon structures. Compared to the other available methods for stress measurement, Raman spectroscopy method is a non-destructive technique, requires minimal sample preparation, and has spatial resolution of less than one micron that makes it suitable for microscale to nanoscale samples. Most importantly, this method is also suitable for the measurement of temperature [4–6] and thermal conductivity [7–11] along with stress. In this research, combined nanoindentation and nanomechanical Raman spectroscopy (NMRS) methods (developed by Gan and Tomar, [12–14]) were used to measure crack tip stress distribution. Through the relation between the Raman shift and stress, the stress or strain can be calculate at each scanning point and then be integrated to show the stress distribution around the area of crack tip. Through the relation between the Raman shift and Raman peak width, the Raman shift caused by the stress and temperature can be separated. As a result, stress distribution around the crack tip at different temperature will be shown. The tested samples used in this research were subjected to 3-point bending based mechanical loading applied by a modified nano-scale loading platform. For the 3-point bending crack tests, the samples were cut into small pieces according to the ASTM D5045 standard with the dimension of 8.8 * 2.0 * 0.5 mm with the length of crack equal to 1.0 mm and the width 0.2 mm. After cutting, the samples were mechanically polished, electrolytically polished and etched. In order to detect the Raman signal, a 1.0 m thick layer of silicon was deposited on the surface of metals that creates negligible Raman shift phenomenon. The schematic diagram of the experimental setup is shown in Fig. 3.1. As mechanical load is applied along one axis, the Raman laser is focused onto the side surface using an objective. Back-scattered Raman signal is collected by the same objective and sent to a spectrometer. Y. Zhang ( ) •V. Tomar School of Aeronautics and Astronautics, Purdue University, West Lafayette, IN, 47907, USA e-mail: yangzhang@purdue.edu; tomar@purdue.edu © The Society for Experimental Mechanics, Inc. 2018 J. Carroll et al. (eds.), Fracture, Fatigue, Failure and Damage Evolution, Volume 7, Conference Proceedings of the Society for Experimental Mechanics Series, DOI 10.1007/978-3-319-62831-8_3 13
14 Y. Zhang and V. Tomar Fig. 3.1 Experimental set-up of nano-mechanical Raman spectroscopy The temperature as well as stress change affect the position of Raman peak. In this work, both full width half maximum (FWHM) and Raman shift were detected and evaluated at different temperatures. The temperature of sample which was obtained from FWHM measurement is used to separate the temperature-induced Raman shift from the stress-induced Raman shift. The overall schedule had three steps for one set of experiments. The first step is to make indentations around the notch tip on the uncoated surface of sample before the bending. These tests are used to determine indentation based stressstrain curve in the area before 3-point bending. Thereafter, while applying load to bend the sample, Raman scanning was performed around the notch tip at the coated surface using the NMRS. After separating temperature effect on Raman shift and combining the stress information from each measuring point, a contour figure of stress distribution around the notch tip can be plotted. Overall 72 indentations (and corresponding Raman spectrums) were scanned to get the mechanical properties mapping, stress distribution and temperature field. For each spectrum, at least 3 repeated tests were performed to eliminate measurement error. After combining all the stress information, contour figures that depicted stress distribution around the notch tip under loads of 500 mN, 2 N and 4 N at room temperature were generated in Fig. 3.2. As shown and discussed before, the stress distribution around the notch tip was not symmetric due to the asymmetry in the material microstructure around the notch tip. Grain boundary and grain structure played an important role in the mechanical property of polycrystalline material. In this research Orientation Image Microscopy (OIM) [15–17] which was an automated indexing of electron back-scattered diffraction (EBSD) pattern was used to obtain the characteristic of microstructure. The microstructural information obtained from the Raman scanned area using OIM was correlated to the stress distribution data that obtained from nano-mechanical Raman spectroscopy. The tip of notch was the stress concentrated area through the figures. Through the comparison of stress distribution mapping for different samples under different loads, it is known that the stress is concentrated and higher at the notch tip as expected for any kinds of microstructure. Additionally, notch tip stress increase with the bending load but not proportional to the load due to the plasticity at the notch tip. Plastic deformation will degrade the strength of material that result in lower stress. Correlation with microstructures of the notch area, it is clear that microstructures play a very important role in the stress distribution. According to the relation between temperature and Raman peak, temperature at every point around the notch tip was measured. After combining all measured temperature information, contour figures that depicted temperature field around notch tip under loads of 500 mN, 2 N and 4 N at the overall temperature of 200 ıC were shown in the Fig. 3.3. The average
3 Characterization of Crack Tip Plasticity in IN-617 Using Indentation and Nano-Mechanical Raman Spectroscopy 15 Fig. 3.2 Stress distribution with microstructures obtained from EBSD of the notch tip under different bending loads of 500 mN, 2 N and 4 N at room temperature temperature of the sample during the test was 200 ıC, which was monitored by thermocouples. However, the temperature along the notch edge was little lower due to heat convection to the ambient air with obvious lower temperature inside the notch. And the difference of temperature at different area was due to difference of thermal conductivity that was affected by the precipitates on the surface, microstructure of surface and the stress conditions. After subtracting the temperature-induced Raman shift contribution, the remaining Raman shift was used to calculate stress. Stress values at every points around the notch tip was measured. Contour plots that depicted stress distribution of around the notch tip under loads of 500 mN, 2 N and 4 N at the overall temperature of 200 ıC were also shown in Fig. 3.3. The temperature field and stress distribution are affected by the microstructure of the notch area and the precipitates. In order to consider microstructure influence one needs to perform crystal plasticity type of simulations. However, dislocation systems data is very limited in this case. Therefore, an alternate approach is pursued in this work. For the purpose of finite element simulations, boundary conditions are shown in Fig. 3.4a. Microstructure of Alloy 617 has average grain size of approximately 150 microns. For each grain using spherical indentation, elastic-plastic stress strain curves were obtained with spacing of 50 microns by converting the experimental load-displacement data [18]. Figure 3.4b shows the
16 Y. Zhang and V. Tomar Fig. 3.3 Stress distribution and temperature field with microstructures obtained from EBSD of the notch tip under different bending loads of 500 mN, 2 N and 4 N at average temperature of 200ıC
3 Characterization of Crack Tip Plasticity in IN-617 Using Indentation and Nano-Mechanical Raman Spectroscopy 17 Fig. 3.4 (a) Overall setup and boundary conditions in FE simulation, (b) indentation stress-strain curve zones around notch area, (c) stress map around notch determined using FE simulation at 500 mN bending load, and (d) stress map around notch calculated theoretically at 500 mN bending load grid used for such calculations. Instead of considering actual grain structures, different elastic and plastic material properties which were obtained from indentation experiments around the notch area were applied to the same area with 68 small rectangular zones of 50 by 50 microns. For each grain average stress-strain behavior was based on spatial average of such curves. These stress-strain curves were fitted to a J2 plasticity model. The validated J2 plasticity model for each grain was taken as a representative for its elastic-plastic deformation behavior. While performing the finite element deformation simulations in Fig. 3.4a, use of the grid shown in Fig. 3.4b leads to the incorporation of averaged microstructure effects and plasticity. In the rest of sample, average material properties that were also obtained from the experiments was applied. From the finite element simulation results shown in Fig. 3.4c it is clear that the average maximum principal stress at the notch tip matches closely with experimental measurements (Fig. 3.2, 500 mN). Difference between the experiments and simulations at other points in microstructure away from the notch can be attributed to the residual stresses due to grain boundaries and precipitates. Magnitude-wise, however, stresses are in the same range in the scanned area using NMRS bringing simulations and experiments close in terms of stress prediction. Theoretical calculations were also performed to predict stress distribution in notch area based on stress intensity factor calculations. The mapping of maximum principal stress obtained in this way around the crack tip is shown in Fig. 3.4d. The difference between the experiments and theoretical calculation can be attributed to crack tip plasticity as well as microstructure effect. However, crack-tip average stresses are in the same range as experiments. In the current research, a novel NMRS measurement platform was used to measure crack (notch) tip stresses duringin-situ mechanical deformation at elevated temperature. Notch tip of sample during 3-point bending tests with an initial notch was scanned. Temperature field and stress distribution in the notch tip area were generated by combining measurement results at each scan point through the relation between the stress, temperature and Raman shift. Based on the nanoindentation stressstrain curve, the yield strength at the notch tip was found to decrease after the 3-point bending. Instead of considering actual grain structures with different material properties, a new FE method was adopted to predict stress distribution around the same notch area as scanning. Predictions from theory and simulations matched closely in stress concentration area near notch with experimental measurements. However, away from notch area deviation in simulation and experimental predictions due to microstructural effects was observed. A significant effect of temperature change induced residual stress and its correlation with microstructure dependent temperature distribution was observed.
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