Rate dependent fracture along a silicon/epoxy interface under mixed-mode loading conditions

Abstract

This paper describes the development of a dual-actuator loading device that was then used to apply asymmetric, transverse end-displacements to laminated beam specimens (silicon/epoxy/silicon) over a range of separation rates. Measurements of the reaction forces, as well as load-point displacements and rotations, were used to determine the normal and tangential components of the crack tip displacements and the corresponding components of the J-integral. This was made possible because the specimens identically satisfied a balance condition. The resulting data set obtained from experiments conducted at five separation rates at each of five mode-mix phase angles is a testimony to the efficiency of the approach. A mixed-mode beam on elastic foundation analysis established that the stiffness of the normal and shear interactions of the silicon/epoxy interface was independent of the separation rate and mode-mix. Furthermore, the stiffness values thus determined were considerably lower than those based on the bulk behavior of the epoxy in tension and shear. The analysis also allowed the crack growth to be tracked in order to establish its onset and the corresponding critical values of the normal and shear components of the J-integral, along with the corresponding strengths and critical crack tip displacements. For each mode-mix, these critical values increased with the separation rate. This increase in properties is in spite of the glassy nature of the bulk epoxy and further suggests the presence of an interphase region in the epoxy adjacent to the silicon. However, the change of mode-mix was accompanied by a change in local separation rates, leading to non-monotonic behavior in the critical J-integral. Following the onset of crack growth, the application of the transverse end-displacements along radial loading paths resulted in simultaneous changes in the local separation rates and mode-mix, implying a fracture criterion that depends on both mode-mix and rate-dependent damage evolution processes.

Introduction

Interfaces abound in many technologically important applications that range from primary structural adhesively bonded joints in aerospace, naval and automotive structures to the multiple interfaces that are common in microelectronics devices and packaging. One potential failure mode in all these heterogeneous systems is interfacial delamination, which can be addressed via fracture mechanics analyses quantifying the strength and durability for design purposes.

Delamination analyses based on interfacial fracture mechanics concepts were pioneered by Williams (1959) and effectively put into practice by Rice, 1988, Hutchinson and Suo, 1992. A striking feature of interfacial fracture mechanics is that the toughness is often a function of the relative amount of tensile and shear tractions on the interface (Chai and Liechti, 1992, Evans et al., 1990, Wang and Suo, 1990) when the crack is constrained to grow along an interface. The toughening of interfaces with increasing shear component has generally been attributed to asperity locking (Evans and Hutchinson, 1989) or increased plastic or viscoplastic dissipation (Chai and Liechti, 1992, Swadener and Liechti, 1998) near the crack front. This so-called linearly elastic fracture mechanics (LEFM) approach is generally sufficient in accounting for the behavior of preexisting flaws as long as the fracture process zone is sufficiently small (Parmigiani and Thouless, 2007, Sills and Thouless, 2013). Alternatively, cohesive zone modeling can accommodate delamination with larger fracture process zones and without the requirement of a preexisting flaw (Mohammed and Liechti, 2000). The ideas behind cohesive zone modeling were originally proposed by Dugdale, 1960, Barenblatt, 1959 in order to mitigate the stress singularities that are the hallmark of LEFM. Since then, it has been applied to interfacial crack growth problems in general (Needleman, 1990), delamination in adhesively bonded joints (Högberg et al., 2007, Li et al., 2006, Sørensen, 2002, Ungsuwarungsri and Knauss, 1987), laminated, fiber-reinforced composite materials (Blackman et al., 2003) and thin films (Shirani and Liechti, 1998) as well as adhesive contact problems (Johnson et al., 1971, Maugis, 1992), as some early examples in an extensive array of literature.

Cohesive zone modeling typically requires a traction-separation relation as the constitutive representation of the interactions between the surfaces. The traction-separation relation for a specific interaction can be determined experimentally by direct or indirect methods (Gowrishankar et al., 2012). The direct method requires two components: the path-independent J-integral and the crack tip displacement, which can be challenging for experiments and may suffer from resolution issues. For materials that are transparent to visible or infrared radiation, crack opening interferometry has been used to characterize the crack tip behavior (Gowrishankar et al., 2012, Mello and Liechti, 2006, Wu et al., 2016). The method can have a resolution of 20 nm and the full crack front can be observed. However, only normal crack tip displacements can be measured.

Several approaches based on digital image correlation (DIC) have been introduced recently. Blaysat et al. (2015) developed a parameter identification approach based on the kinematics of double cantilever beam specimens and concepts from integrated DIC, where the unknown degrees of freedom are the properties that define a traction-separation relation, rather than displacements and rotations, thereby increasing computational efficiency and robustness. Gorman and Thouless (2019) conducted an extensive study of the use of DIC for tracking the evolution of the cohesive zone and extracting traction-separation relations. In a study of the rate dependence of fracture between polymer modified bitumen and aluminum under mode I loading (Rajan et al., 2018), two stereo DIC systems were used. One, operating at higher magnification, was focused on the crack tip region to measure the crack tip opening displacement with a resolution of approximately 40 nm. The low magnification system was used to determine the strains ahead of the crack tip as well as the load line displacement. The same group has recently extracted the mode I and mode II traction-separation relations of uncured thermoset tows using a rigid double cantilever beam arrangement in combination with DIC (Rajan et al., 2020).

In the current paper, we investigate the effects of rate and mode-mix on fracture of an interface by following the direct method (Ouyang and Li, 2009, Wu et al., 2019), which significantly simplified the J-integral concept for mixed-mode fracture and allowed the crack tip displacements to be determined from the remote measurements at the loading point. The cohesive traction at the interface is then attained as the derivative of the J-integral with respect to the local separation (Sorensen and Jacobsen (2003)), under the assumption that the same traction-separation relations are followed along the interface. Wu et al. (2019) utilized the reflection of a laser beam on a 45° mirror to measure the end rotations of end-loaded split (ELS) and end-notched flexure (ENF) specimens, which offered a resolution of ~10⁻⁴ rad. However, the range of the beam deflection was restricted by the size of the position sensing detector, which was used to receive the reflection beam signal. Here, we have found that DIC is simpler and more robust in measuring the end displacements and rotations. It provides the trajectory of the beam deflection all the way to the crack tip in order to determine the normal and tangential components of the J-integral and crack tip displacements. It also allows the mode-mix to be controlled more easily than using a selection of asymmetric specimens.

Many test methods have been successfully developed for characterizing fracture at interfaces under all three fracture modes and combinations thereof. The most commonly used type of specimen for determining the fracture toughness of a bi-material system is the double cantilever beam specimen as established by Kanninen (1973). As a logical evolution from quasi-static testing with double cantilever beam specimens, alternative specimen geometries, such as reinforced adherends (Jain et al., 1998, Marzi et al., 2014) and tapered adherends (Brussat et al., 1977), were studied for measuring the mode I fracture toughness at high separation rates. Mixed-mode interactions have been characterized by introducing asymmetries either in the specimen geometry and materials or loading conditions. Examples of mixed-mode fracture tests with asymmetry in the specimen geometry and materials include the asymmetric double cantilever beam (Sundararaman and Davidson, 1997, Xiao et al., 1993), four-point flexure and composite cylinder (Cao and Evans, 1989, Charalambides et al., 1990), the end loaded split (Hutchinson and Suo, 1992, Wang and Vu-Khanh, 1996), and the compact tension shear specimens (Mahajan and Ravi-Chandar, 1989). Most of these previous works focused on the effect of mode-mix on the fracture toughness following the LEFM approach. It remains a challenge to characterize the mixed-mode traction-separation relations for an interface (Sorensen and Jacobsen, 2003). A direct method was proposed by Wu et al. (2016) to determine mixed-mode traction–separation relations based on a combination of global and local measurements using the end loaded split (ELS) configuration for a silicon-epoxy interface, where the epoxy thickness was varied to obtain phase angles ranging from −42° to 0°. More recently, mixed-mode traction-separation relations were extracted directly for a silicon/epoxy interface with phase angles ranging from −53° to 87.5°, using asymmetrical end-notch flexure and end-loaded split specimens with different adherend materials (Wu et al., 2019).

Asymmetry can also be introduced by applying uneven loads to a symmetric laminate. Reeder and Crews (1990) pursued such an approach to study mixed-mode fracture in laminated fiber reinforced composites. Similarly, Fernlund and Spelt (1994) developed a complex loading jig consisting of a linkage system which induced an asymmetry in the forces acting on the upper and lower adherends. Davidson and Sediles (2011) went a step further by developing a device that made use of bending and torsion on a laminate to produce all three fracture modes. Although none of these devices were used to extract traction-separation relations, there were some questions from a theoretical standpoint (Suo et al., 1992) as to the suitability of applying uneven end loads rather than moments for providing fracture properties under large scale bridging conditions. Because configurations that employ uneven end moments (Jacobsen and Sørensen, 2001, Lindhagen and Berglund, 2000, Sørensen and Jacobsen, 2009, Sørensen and Jacobsen, 1998) provide crack tip stress fields that are invariant with crack length, it was postulated that they would provide true material properties, uninfluenced by structural effects. This point has recently been addressed by Pappas and Botsis (2019), who also developed a more convenient way to apply uneven bending moments. They found that applying uneven end loads or moments to an adhesively bonded laminate resulted in very similar traction-separation relations even though the damage zone was relatively large. On the other hand, the same was not true of a laminated fiber reinforced polymer with a large and complex bridging zone.

All the approaches for controlling mode-mix that have been discussed so far are inherently proportional loading devices, which do not allow the effects of more complex mixed-mode loading paths to be followed. Biaxial loading devices (Chai and Liechti, 1991, Liechti and Knauss, 1982, Mello and Liechti, 2004) that apply uniform tension and shear to a bimaterial strip are certainly capable of achieving such a goal, but are complex. A conceptually simpler approach was provided by Singh et al. (2010) who used a dual actuator device to apply uneven end loads to a symmetric laminate. The approach taken here mirrors their approach, albeit with simpler actuation due to the relatively low loads that were required here. The device has recently been used to study the rate-dependent fracture of a silicon/epoxy interface under the nominally mode-I condition with symmetric loading (Yang et al., 2020).

The remainder of this paper is organized as follows. In section 2, we describe the configurations of the experimental setup and the specimen preparations in detail. Then we present a series of analyses in section 3 concerning the design of the experiments, the mode-mix analysis, and the post processing of the raw data for extraction of the mixed-mode properties. Next, a complete set of results are displayed in section 4 where the effects of the separation rate and mode-mix are discussed at length. Conclusions are provided in section 5.