Rashid K. Abu Al-Rub 1 e-mail: rabualrub@civil.tamu.edu Abu N. M. Faruk Zachry Department of Civil Engineering, Texas A&M University, College Station, TX 77843 Coupled Interfacial Energy and Temperature Effects on Size-Dependent Yield Strength and Strain Hardening of Small Metallic Volumes Plasticity in heterogeneous metallic materials with small volumes is governed by the interactions of dislocations at interfaces. In particular, interfaces of a material confined in a small volume can strongly affect the mechanical properties of micro and nanosys- tems. In this paper, the framework of higher-order strain gradient plasticity theory with interfacial energy effect is used to investigate the coupling of interfacial energy with temperature and how it affects the initial yield strength (i.e., onset of plasticity) and the strain hardening rates of confined small metallic volumes. It is postulated that the inter- facial energy decreases as temperature increases such that size effect decreases as tem- perature increases. As an application, the size effect of thermal loading of a film- substrate system is investigated. It is shown that the temperature at which the film starts to yield plastically is size-dependent, which is attributed to the size-dependent yield strength. Furthermore, the flow stress is more temperature sensitive as the size decreases. DOI: 10.1115/1.4002651 Keywords: size effect, interfaces, surface energy, temperature, thin film 1 Introduction Plasticity in heterogeneous materials with small volumes e.g., thin films, thin wires, grains in polycrystals, and particles in com- posites is governed by the interactions of dislocations at inter- faces e.g., thin film-substrate interface, grain boundary, and particle-matrix interface. These include interactions of existing dislocations, as well as the nucleation of dislocations at an inter- face. The rational for interface dominated plasticity is simple: Dis- locations glide through the single crystal domain with relative ease but pile-up at interfaces so that interface interactions become a critical step in continuing plastic deformation. While the details of dislocation interactions at interfaces take place at the atomic scale 1, and the behavior of dislocations in bulk is most accu- rately modeled by discrete dislocation dynamics e.g., Refs. 2–4, both of these models are much too expensive and imprac- tical for analyzing the resulting bulk behavior. The need for a continuum but nonclassical framework for describing the plas- ticity across interfaces including the prediction of size effects seen in many experiments e.g., Refs. 5–12 is necessary. The interfacial interactions can be characterized within the framework of higher-order not lower-order nonlocal strain gra- dient plasticity theory e.g., Refs. 13–16 through a proper inter- pretation of the physical nature of the higher-order interfacial boundary conditions that result from the application of the prin- ciple of virtual power or virtual work. Recently, initial attempts have been made to relate these nonstandard boundary conditions to the interfacial energy at interfaces and found out that this proposition can be used successfully in predicting the increase in the initial yield strength, flow stress, and strain hardening rates with decreasing size in micro and nanostructured materials 16–19. Abu Al-Rub 19 investigated different mathematical forms of the interfacial energy and formulated, besides the yield condition for the bulk, a yieldlike condition for the interface. However, to the authors’ best knowledge, the effect of temperature on the interfacial energy and on the size effect of small-scale metallic volumes has not been investigated. Therefore, the objec- tive of this paper is to utilize the framework of higher-order strain gradient plasticity theory, as formulated in Refs. 16,19, to inves- tigate the coupling of interfacial energy with temperature and how it affects the initial yield strength i.e., onset of plasticity and the strain hardening rates of confined small metallic volumes. Particu- larly, the effect of cooling of a metallic thin film on a silicon substrate is investigated. It is shown that the cooling temperature at which the film starts to yield is size-dependent such that as the film thickness decreases, more cooling is needed to yield the film plastically. This paper is organized as follows. The framework of higher- order strain gradient plasticity theory with the consideration of interfacial energy effects is outlined in Sec. 2. The temperature and functional dependence of the interfacial energy on the inter- facial plastic strain is discussed in Sec. 3. In Sec. 4, the presented higher-order gradient plasticity is applied to the thermal cooling of a metallic thin film on a silicon substrate. Conclusions are out- lined in Sec. 5. 2 Higher-Order Strain Gradient Plasticity With Inter- facial Effects In order to be able to model the small-scale phenomena, such as the effect of size of microstructural features on the material me- chanical properties, the higher-order strain gradient plasticity theory that can predict the size scale effects is considered here. In this section, the recently developed thermodynamic framework for higher-order gradient plasticity theory by Abu Al-Rub et al. 16 is utilized to derive the necessary constitutive equations. 1 Corresponding author. Contributed by the Materials Division of ASME for publication in the JOURNAL OF ENGINEERING MATERIALS AND TECHNOLOGY. Manuscript received March 17, 2010; final manuscript received August 12, 2010; published online December 3, 2010. Assoc. Editor: Assimina Pelegri. Journal of Engineering Materials and Technology JANUARY 2011, Vol. 133 / 011017-1 Copyright © 2011 by ASME Downloaded 05 Dec 2010 to 165.91.74.118. Redistribution subject to ASME license or copyright; see http://www.asme.org/terms/Terms_Use.cfm