Introduction

Moiré interferometry [1] is a well established optical method for the measurement of in-plane displacements on the surface of solids. It utilizes coherent light (typically lasers) to form an interferometer (optical assembly) and a diffraction grating replicated on the surface of the subject. The interferometer directs two pairs of collimated wave-fronts, a pair for each orthogonal direction, to the specimen diffraction grating at specific angles. When the specimen and thus grating deform, the diffracted wavefronts interfere to produce fringe patterns that represent the in-plane displacements on the surface of the sample. Subsequent analysis of the two displacement fields using strain displacement relationships provides the strain field. Modern methods of phase-shifting are often employed to automate the fringe analysis procedure and help define the sign of the displacement and strain information. For the most part, the technique is limited to flat surfaces, although provisions for singly curved surfaces have been made. Additionally, the size of the area of interest is typically on the order of 5 cm by 5 cm or smaller. Recent studies using microscopes have made it possible to achieve a viewing area as small as 200 μm × 200 μm. The technique has been thoroughly documented in numerous texts and papers that describe the history, theory, implementation, and application. This paper will only cover recent applications (from the late 1990s to the present).

In an article titled “Perspectives in Experimental Solid Mechanics”, published in the International Journal of Solids and Structures, in 2000, W. Knause [2] reviewed the significance of experimental methods for the advancement of the understanding of mechanical phenomena in solids. In that article he singled out moiré interferometry as a tool that has both high measurement and spatial resolution. The following quote is taken from that paper. “Perhaps the interferometric and moiré interferometric methods deserve special attention because of their power to resolve displacements measured in terms of the wavelength of light (~ one micron) and because of their potential (when used carefully) for high spatial resolution of the displacement field. These two methods also illustrate the evolution of experimental methods over the last 100 years and thus demonstrate how the need in a certain science field (solid mechanics) culls a new method (moiré interferometry) from a well established physical principle for a special application.” He goes on to say, “Because of this high resolution power of moiré interferometry, it has been an important addition to the repertoire of tools for experiments and became thus a favorite tool for refined deformation measurements on electronic micro chips; in this connection this method has served virtually the same purpose for these small devices as photoelasticity has for the larger engineering structures during the middle of this century; the major difference being that photoelasticity addresses the stress state more directly than moiré interferometry, which renders displacements.”

This statement summarizes the state of moiré interferometry contributions at the beginning of the last decade. Since then, moiré has continued to be an invaluable tool in our quest to understand mechanical phenomena.

The theory of the moiré interferometry technique can be found in numerous publications including those written by groups led by Post [1, 3–5], Mckelvie [6–9], Walker [10–12], Cloud [13], Dai [14–17], Morimoto [18, 19] and the current contributing authors. In this text we will expand on recent applications in the research on microelectronics devices, material characterization, micromechanics, residual stress, composite materials, fracture mechanics, and biomechanics. In the short format associated with Experimental Mechanics it is impossible to cover all applications. The ones covered represent only a fraction and the authors apologize for those omitted in this text.

Microelectronics Device

Microelectronics devices contain many electronic components within an active silicon chip, such as transistors, capacitors, resistors, etc. To form a usable device, a silicon chip requires protection from the environment as well as both electrical and mechanical connections to the surrounding components. The various conducting and insulating materials involved in the devices have different coefficients of thermal expansion (CTE). When the chip is powered up and thus the device is subjected to a temperature excursion, each material expands at a different rate. This non-uniform CTE distribution produces thermally induced mechanical stresses within the device assembly. As the components and structures involved in high-end microelectronics devices are made smaller, the thermal gradient increases and the strain concentrations become more serious. Hence, there is a continuously increasing activity in experimental analysis, both for specific studies and for guidance of numerical analyses. Moiré interferometry has been taking a leadership role for experimental analyses since Guo and his co-workers introduced a special technique to apply gratings on specimens with a complex geometry [20].

A special technique is required for the cross sections of microelectronics devices because they usually have such a complex geometry that the excess epoxy produced by the usual grating replication procedure cannot be removed [20, 21]. The excess epoxy is critical since it could reinforce the specimen and change the local strain distribution. An effective replication technique was developed to circumvent the problem. First, a tiny amount of liquid epoxy is dropped onto the grating mold; the viscosity of the epoxy should be extremely low at the replication temperature. Then, a lintless optical tissue (a lens tissue) is dragged over the surface of the mold, as illustrated in Fig. 1. The tissue spreads the epoxy to produce a very thin layer of epoxy on the mold. The specimen is pressed gently into the epoxy, and it is pried off after the epoxy has polymerized. Before polymerization, the surface tension of the epoxy pulls the excess epoxy away from the edges of the specimen. The result is a specimen grating with a very clean edge.

Fig. 1
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Replication procedure for small specimens of complex geometry

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The early applications of moire interferometry used a bi-thermal loading [22]. In this technique, the specimen grating is applied at an elevated temperature, and then the specimen is allowed to cool to room temperature before it is observed in the moiré interferometer. Thus, the deformation incurred by the temperature increment is locked into the grating and recorded at room temperature. Numerous examples can be found in the literature [20, 21, 23–41].

An example is illustrated in Fig. 2 [24, 35, 41]. The specimen is a flip-chip plastic ball grid array (FC-PBGA) package assembly. In the assembly, a silicon chip (6.8 mm × 6.1 mm × 1.2 mm) was first attached to an organic substrate through tiny solder bumps called C4 interconnections. The gap between the chip and the substrate was filled with an epoxy underfill to reduce the strains of solder bumps. This subassembly was then surface-mounted to a typical FR-4 printed circuit board (PCB) through larger solder ball arrays to form a final assembly. The assembly was cut and its cross-section was ground to produce a flat, smooth, cross-sectional surface. The specimen grating was replicated at 82°C and the fringes were recorded at room temperature (ΔT = −60°C). Very clean edges of the specimen grating are evident. The substrate had a higher coefficient of thermal expansion (CTE) than the PCB. The substrate contracted more than the PCB during cooling, while the deformation of the substrate covered by the chip was constrained by the low CTE of the chip. This complicated loading condition produced an uneven curvature of the substrate. The substrate was connected to the PCB through the solder balls and the difference of curvature between the substrate and the PCB was accommodated by the deformation of the solder balls. The experimental evidence provided by moiré interferometry was essential to revealing important design parameters.

Fig. 2
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U and V displacement fields of FC-PBGA package assembly, induced by a bi-thermal loading of ΔT = −60°C

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The bi-thermal loading was extended to cyclic loadings, where the package assemblies were subjected to thermal cycles or mechanical cycles first and residual deformations were documented at room temperature. Han [27] determined inelastic damage accumulations in solder columns of a ceramic column grid array assembly after subjecting it to thermal cycles. Later, Basaran et al. [42], Liu et al. [43] and Tunga and Sitararman [44] used a similar procedure for BGA assemblies but at the much higher number of cycles, and Kwon et al. [45] for an assembly using anisotropic conductive adhesive. Zhao and Basaran also measured the inelastic deformations of solder joints, caused by vibrations and shocks [46].

Thermal-cycling induced residual deformations documented by moire interferometry are illustrated in Fig. 3 [27]. The specimen was a ceramic column grid package assembly [Fig. 3(a)] which was subjected to four thermal cycles of 125°C and −40°C. The residual U and V displacement fields of the column are shown in (b), which were used to determine the deformed shape of the column (c). After thermal cycles, the column surrounded by the eutectic solder fillet experienced much more severe permanent deformation, which increased the bending deformation at the top. This effect was confirmed by the results from an actual accelerated thermal cycling (ATC) test.

Fig. 3
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Column grid package (a) assembly; (b) U and V displacements and (c) exaggerated deformed shape of solder column after four thermal cycles, and (d) failed column after the actual ATC test

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The applications were further extended into the domain of real-time observation under a mechanical loading [47–49]. Stout et al. measured deformations of the PBGA assembly subjected to four-point bending [47]. Later Wang et al. [48] used four-point bending to investigate the effect of packaging on the Cu/Low-k interconnects, and Joo et al. [49] also used four-point bending to investigate different deformation modes caused by the mechanical bending while comparing them with the thermo-mechanical deformation modes. Figure 4(a) shows schematically a flexure test apparatus (the insert shows the positions of specimen and loading pins) used in [49]. The mechanical loading induced deformation (b) shows a symmetric deformation causing the maximum deformation in the outmost solder ball, while the thermally-induced deformation (c) caused the maximum deformation at the solder ball directly under the edge of the chip.

Fig. 4
figure 4

(a) Schematic diagram of a flexure test apparatus (the insert shows the positions of specimen and loading pins). Representative U field fringe patterns produced (b) by mechanical bending of 28.2 N∙mm and (c) by thermal loading by heating to 80°C (Courtesy of J. Joo, Chungbuk National University)

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More recently, robust schemes of moire interferometry became available for real-time observations of thermally-induced deformations, most notably PEMI (Portable Engineering Moiré Interferometer). They were employed to study the temperature-dependent thermo-mechanical behavior. The schemes were implemented with convection-type or conduction-type environmental chambers that provide the temperature control required in accelerated thermal cycling (isothermal loading) [35, 41, 50–65]. Park et al. [66] and Yang et al. [67] tested package assemblies under power cycling by heating the chip directly. Figure 5 shows the schematic of the experimental setup of the power cycling and U field moiré fringes obtained during the cycling [66].

Fig. 5
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Schematic of the experimental setup of power cycling and U field moire fringes at two different times (Courtesy of S. B. Park, SUNY Binghamton)

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Other loading conditions were also used to measure deformations. Ye et al. [68, 69] measured the in situ displacement evolution of lead-free solder joints under electric current stressing. Stellrecht et al. [70, 71] developed an effective scheme to measure deformations induced only by hygroscopic swelling. Later Tsai et al. [72] and Park et al. [73] used a similar procedure to measure hygroscopic deformations of a plastic package and a conductive adhesive package, respectively. Figure 6 illustrates the deformation induced by hygroscopic swelling, where fringe patterns were obtained at 85°C after subjecting a plastic package to 85°C/85%RH [71].

Fig. 6
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Fringe patterns obtained at 85°C after complete moisture absorption (Courtesy of E. Stellrecht, Sierra Research)

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Material Characterization

Traditionally, material property characterization is performed using standard test methods and conventional strain measurement techniques such as strain gages and extensometers. With the ever increasing complexity of material behavior, comes the need for more advanced methods for characterizing then. For instance active materials, or smart materials, undergo phase changes that can be complicated by hysteresis, and thermal effects. Full-field methods can provide valuable information that can be used to understand the physics of how such materials behave and thus help to build and validate numerical models.

Perry et al. [74] used moiré interferometry to study the deformation in SE-508 seamless drawn tubing heat treated at 500 C. Figure 7 shows a sequence of wrapped fringe patterns during the tensile loading of the material at a temperature above the austenite finishing temperature. Transformation to martensite initiates at the top and bottom of the gage section and works its way towards the center of the specimen. From this test the stress versus strain behavior from both a global measurement and the local measurement, taken from the moiré fringe patterns could be extracted from the sample.

Fig. 7
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The vertical displacement field for a Nitinol tensile specimen undergoing phase transformation (Courtesy of Perry)

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The documentation of material heterogeneity is enhanced by the use of full-field methods. Guo et al. [75] recently reported deformation of polycrystalline aluminum alloys using moiré interferometry. They studied a specimen whose grain size was on the same scale as the width of the tensile specimen that was used for loading. They were able to extract the strains from individual grains and build a global and local stress-strain response. Figure 8 shows the region of the specimen that was studied, and the horizontal and vertical displacement fields. The grain boundaries are visible and the heterogeneous behavior can be seen in the fringe patterns.

Fig. 8
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The deformation heterogeneity can be seen in the moiré interferometry fringe patterns applied to polycrystalline aluminum alloy (Courtesy of Guo)

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For simple tensile tests on isotropic materials, the stress-strain curve and Poisson’s ratio can be extracted using conventional strain measurement approaches. These two constants are the only ones needed to characterize the linear stress-strain behavior. For anisotropic materials, there are more constants, and as such, more tests are required to extract the constants thus adding time and expense to the process of characterization. As a result, there has been a push to employ a single specimen of complex geometry, full-field measurements, and inverse methods to determine multiple material properties in orthotropic materials. In 2004 Wang et al. [76] used moiré interferometry in combination with a circular disk, loaded in diametrical compression, to extract material properties from an isotropic material to demonstrate the potential of the inverse method approach. Figure 9 shows the loading condition, the theoretical displacement field (horizontal field in this case), a wrapped moiré displacement field and an unwrapped displacement field. They used least squares to evaluate the differences in the model and experiment. It was shown that the method could determine the modulus of elasticity and the Poisson ratio to within 3% of the handbook value.

Fig. 9
figure 9

(a) Circular disk in diametrical compression, (b) numerical model of horizontal displacements with added noise, (c) wrapped moiré interferometry fringe pattern and (d) unwrapped horizontal displacement pattern (Courtesy of Z. Wang, Catholic University)

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An open-hole tension specimen was employed by Molimard et al. [77] to measure four orthotropic plate constants from a single geometry. They used phase-shifted moiré interferometry to measure full-field surface displacement and strain. Figure 10 shows the geometry and loading fixtures used in their tests. Displacement and strain contours were measured in the area around the open-hole. The principal employed was to minimize the discrepancy between experimental and theoretical strain results using a Levenberg-Marquardt algorithm. Comparisons between the experimental and analytical shearing strain values are presented in Fig. 10. The method takes into consideration the optical system, signal processing, and the mechanical aspects. Cost functions were investigated leading to a simple mathematical form. Two models were used: an analytical model based on the Lekhnitskii approach and the finite element method. The researchers were able to identify the four elastic constants to within 6% of those measured using traditional means.

Fig. 10
figure 10

Open Hole specimen and loading configuration with experimental shear strain distribution, analytic shear strain distribution, and differences after the application of difference minimization using the Levenberg-Marquardt algorithm (Courtesy of Molimard)

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Aside from mechanical properties such as modulus of elasticity and Poisson’s ratio, there are other material properties that are of interest to engineers. When utilizing concrete, for instance, the shrinkage during cure as a function of composition, time, humidity, and temperature plays a vital role in the ultimate strength and integrity (free from cracks). There have been numerous methods developed to understand this complex phenomena. Recently, Chen et al. used moiré interferometry and inverse methods to determine shrinkage coefficients in cement, mortar and concrete materials [78]. In Fig. 11 the shrinkage of cement can be seen as a function of time at various humidities. The technique utilizes the methodology of the cure reference method, formerly developed for composite residual stresses by attaching a diffraction grating on the cement sample during the initial solidification of the cement. The grating then acts as a datum from which subsequent deformations were measured over a period of 7 days. The method was employed in parallel with an axi-symmetric FEM model that modeled the grating as a semi-permeable membrane.

Fig. 11
figure 11

The average shrinkage across the width of a cement specimen versus time (days) for various combinations humidities. The fringe patterns show the U and V displacement fields for the first 3 days (Courtesy of Chen)

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Micromechanics

Many fields of study require deformation measurements of tiny specimens or tiny regions of larger specimens. The mechanics of microelectronic assemblies is an example, where the ever-increasing demand for closer packing exacerbates the problems of thermal stresses. Other fields include crack-tip analyses in fracture mechanics; grain and intragranular deformations of metals and ceramics; fiber/matrix interactions in fiber reinforced composites; interface problems; etc.

The small size and the need for high spatial resolution require microscopic viewing of the specimen. Microscopic moiré refers to this special implementation of moire interferometry for microscopic viewing [79, 80]. Tiny interferometers were fabricated to accommodate microscope objectives as an imaging system for large deformations (Shield and Kim [79]) and small deformations (Han and Post [80]).

In the approach proposed by Han and Post, an “immersion interferometer” was developed, whereby the specimen is coupled optically to the interferometer by a thin layer of immersion fluid to reduce the wavelength of the light propagating in the medium (Fig. 12) and thus increased the upper sensitivity limit of moire interferometry. Later Liou and Prakash [81] proposed a microscopic moiré approach using a transmission diffraction grating that allows a simple and quick change of the virtual reference grating vector without disturbing the optical alignment of the other components in the optical train.

Fig. 12
figure 12

Optical paths in an immersion interferometer and arrangement for U and V fields in microscopic moiré interferometry

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Within a small field of view, the relative displacements are typically small unless the strains are extremely large. For those applications, displacement resolutions were increased by digital image processing techniques: the optical/digital fringe multiplication (O/DFM) method [80, 82] and the phase shifting technique [83].

Bastawros and Kim [84] utilized the approach proposed by Shield and Kim to measure the in-plane components of the Almansi strain tensor. La Porta et al. [83] used microscopic moiré interferometry with the phase shifting technique to investigate the near-tip fields of a precracked stainless steel specimen under load. The approach proposed by Han and Post was combined with the O/DFM method to document the heterogeneous deformation of titanium in elastic tension [85], the fiber/matrix interactions in a unidirectional Boron/Aluminum metal-matrix composite [86], micromechanical thermal deformations of semiconductor packages and subassemblies [30, 41] and the changes in microstrain across bonded dentin interfaces [87]. Figure 13 shows the microscopic displacement fields around two plated-through-holes (PTH), induced by ΔT of −80°C, where the contour interval is 104 nm/fringe [86]. The fringe patterns clearly show the homogenous nature of the plug material inside the PTH and the heterogeneous nature of fiber/resin laminated areas located between the PTHs.

Fig. 13
figure 13

Local CTE variations around plated-through-hole, induced by ΔT of −80°C. The contour interval is 104 nm/fringe (Courtesy of S. Cho, Intel)

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Residual Stress Measurements

In recent years moiré interferometry has been an ever-increasing choice for the investigation of residual stresses in a wide variety of materials, including both homogeneous, isotropic materials and heterogeneous, anisotropic materials, such as fiber reinforced composites. Residual stresses in isotropic materials are induced by non-uniform cooling upon solidification, gradients in material properties, stress history and shot-peening. These stresses can be deleterious to the ultimate load caring capacity of a structure or can be used to suppress tensile stresses and thus they can be advantageous. In composites, residual stresses are produced by the difference in the coefficient of thermal expansion of the constituents (matrix and reinforcement) and the chemical shrinkage of the matrix material during polymerization. In most cases these stresses detract from the ultimate load capacity of the material.

Aside from the full-field nature, high spatial and measurement resolution of moiré interferometry there are other characteristics that make the technique well suited for residual stress measurements. The process of applying the grating to the specimen acts to establish a datum from which subsequent deformation can be referenced. By retaining the grating mold this reference can be recovered through the process of tuning the interferometer to match the grating mold. Subsequent deformations of the specimen grating caused by tractions, temperature change, moisture absorption, chemical shrinkage and machining to release internal stresses can be referenced back to the original undeformed grating. This characteristic has been exploited for hole-drilling, sectioning and composite cure process documentation.

Hole-drilling

The hole-drilling method was originally developed for anisotropic materials by Mathar [88] in 1934. The method relieves stresses via the creation of a free surface (hole surface) and produces a stress/strain gradient in the surrounding material. Strains are traditionally measured with special strain gages [89, 90]. This can be limiting because the strain gradients are not captured by the gage since it only provides average strain values over the area of the gage. As a result, many researchers have utilized full-field methods such as brittle coatings [91], photoelastic coatings [92], holographic interferometry [93], electronic speckle pattern interferometry [13], shearography [94], interferometric strain gages [95] and digital image correlation [96]. In 2010 Schajer [97] wrote a review article on advances in hole-drilling while independently Nelson [98] wrote a review on the determination of residual stresses using optical methods. Both of these, quite recent articles list moiré interferometry as a viable technique that provides good spatial and measurement resolution, can determine the displacements at the hole edge, but can be cumbersome because of grating transfer.

Moiré interferometry was first used for hole-drilling by Nicoletto in 1988 and reported in 1991 [99]. He was able to measure the residual stress in areas dominated by a gradient. Recently, others have utilized the moiré/hole-drilling method for a variety of materials [100–107]. Advances in the technique include the use of automated fringe analysis through the use of fringe shifting, and incremental hole-drilling in combination with moiré. Jian Lu’s research group at the University of Technology in Troyes France (UTT) regularly collaborated with numerous practitioners of moiré including F.L. Dai and both of the contributing authors to extend the method of hole-drilling moiré interferometry to new levels. With Han’s input the incremental hole-drilling method was established and applied to isotropic materials. In a two part paper, Wu, Lu and Han [104, 105] describe the process and the theoretical analysis methods to determine the stresses from the strain information. Wu et al. [106] then extended the method to measure residual stresses on the ply level of composites. Using a 2 mm diameter drill bit and computer controlled drilling increments coincident with the interfaces between plies, displacement fields were recorded through the thickness of the composite. An expression describing the relationship between displacements on the surface, stresses in each layer, in-plane direction cosines, and a set of coefficients was employed. A rigorous calibration of the constants was performed using the full-field, moiré displacement fields and a 3-D finite element model of the 16 ply composite. Figure 14 shows two sets of moiré displacement patterns around the hole after the mill had drilled to two depths (h in the figure). The investigators were able to determine the residual stresses in each ply of a [02/902]2s laminate.

Fig. 14
figure 14

Moiré fringe patterns for two drill depths, the finite element model used for calibrating coefficients, and the residual stresses as a function of thickness (Courtesy of Wu)

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In order to automate fringe analysis, phase shifting algorithms have been employed with the hole-drilling method. An example of this can be seen in the study conducted by Ya et al. [101]. Figure 15 shows the resulting wrapped and unwrapped phase distributions extracted from the fringe patterns for a study conducted on shot-peened aluminum.

Fig. 15
figure 15

Wrapped phase distributions for the (a) U field (b) V field and phase unwrapped distributions for the (c) U field and (d) V field. Residual stresses as a function of depth are presented for a shot-peened aluminum specimen (Courtesy of Ya)

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Composite Sectioning

Much like the hole-drilling method, by sectioning rather than drilling, a free surface is created thus leading to stress/strain redistribution. Whenever a multidirectional laminate is cut, a free edge is created and residual stresses are liberated. Stresses at the free edge must satisfy equilibrium and thus the normal stress perpendicular to, and the shear stress components parallel to the newly developed surface must be zero. Stresses still exist in the interior of the composite, and thus there is a gradient established by forming the free edge. One can measure the strain relief as a result of the creation of the free edge and then relate this to the stresses that existed in the laminate before the cut. There is an underlying assumption that the cutting method itself does not create a deformation field (i.e., no plastic deformation). For most fiber reinforced polymer materials, a well lubricated, slow speed, diamond impregnated cut off wheel has been shown to produce negligible local deformation. This can be verified by cutting a unidirectional composite, where ply level residual stresses do not exist.

A number of researchers have utilized this method to determine the residual stress in laminates [108–112]. Some notable work on the topic was performed by Joh, Gascoigne, Lee and Czarnek. Recently Schoeppner et al. [112] investigated the residual stresses and strain in a composite bonded joint. The study was aimed at validating a polynomial spline displacement approximation method using a thermomechanical linearly elastic analysis. Laminated specimens were bonded together at an elevated temperature and then a diffraction grating was applied to the edge of the composite at room temperature. A cut was then made to liberate stresses. Moiré interferometry was then used to acquire the displacement fields. Fringe shifting was used to determine strain fields and these were compared to the model. Results can be seen in Fig. 16.

Fig. 16
figure 16

The vertical strains εz, from the numerical model and moiré interferometry, on the surface of a bonded joint are compared (Courtesy of Schoeppner)

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Cure Reference Method

The cure reference method [113, 114] introduced in 1999 utilizes moiré interferometry and the application of a diffraction grating on the surface of a composite during the autoclave curing process, as shown in Fig. 17. This grating forms a datum from which subsequent thermal stresses can be referenced. Additionally, the method is capable of measuring the combination of the thermally induced and chemically induced components of strain. Unlike many of the former methods, it is capable of determining the residual stresses on any laminate stacking sequence (not just cross-plies), although standard Kirchoff assumptions are made in the analysis (therefore variations due to through-thickness cure gradients cannot be measured). If the specimen is brought to the cure temperature and the strain is monitored, the thermal and chemical contributions can be separated. By applying the cure reference method to a unidirectional material, the “free residual strain”, defined by the thermal expansion and chemical shrinkage terms combined, can be measured. Then, by applying the cure reference method to a laminate (in the same autoclave cycle) the stresses can be calculated on the ply-scale from the laminate strain information and the free residual strains, within the context of laminate theory.

Fig. 17
figure 17

The cure reference method and fringe patterns of displacement on the surface of a laminated composite

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Experiments on the X-33 reusable launch vehicle laminate were conducted at the University of Florida using a combination of the cure reference method and strain gages. The strain on the surface of multidirectional and unidirectional composites was measured through the temperature range from cure to liquid Nitrogen. It was determined that approximately 20% of the strain at cryogenic temperatures originates from chemical shrinkage and the remainder from a thermal expansion mismatch. A series of experiments was performed to determine the dependence of the coefficient of thermal expansion α2, the modulus E2 and the shear modulus G12 on temperature (these three properties are highly dependent on temperature while the others are nearly independent of temperature). These temperature dependent material properties were then utilized in the analysis. It was shown that the laminate configuration that was presumed to be within a safe operating condition at cryogenic temperature, with a predicted factor of safety of 1.3, actually had a safety factor of 0.8, once the temperature dependent properties and chemical shrinkage residual stresses were incorporated in the analysis. Unfortunately, this analysis was performed after the failure of the fuel tank and the cancellation of the entire X-33 program. In a parallel optimization study, an alternative laminate sequence was developed in order to carry the applied loads, and at the same time, resist residual stress failure. The resulting angle-ply laminate, [±25]n was tested using the cure reference method in parallel to the X-33 laminate. It was found that it retained a safety factor of 1.8 even when the chemical shrinkage term and the temperature dependent material properties were used in the analysis.

Composite Materials

Through the 1980’s and 1990’s composite applications represented a large portion of the studies that utilized moiré interferometry. Composites pose a significant challenge because they are heterogeneous on the fiber-scale, ply-scale and in some cases have complex textile architectures that dominate the strain field; therefore high spatial resolution is required to understand their mechanics. Additionally, most of the high performance material systems, such as carbon fiber/epoxy matrix, are quite stiff and the strain range is typically below 1% requiring a technique with high measurement resolution. Moiré interferometry is well suited to make measurements on the ply-scale and through the entire strain range of most composites; therefore it continues to be a valuable experimental technique for composites applications. In this section we will review a number of recent applications that highlight the capabilities of the technique.

Textile Composites

Textile composites utilize reinforcement in the form of a fabric that can be woven, braided, knit or stitched and then combined with matrix materials, via resin transfer techniques, hand lay-up, or prepreg methods. Textiles can have distinct advantages over traditional laminates made from unidirectional layers, including better conformability over complex 3-D tools, enhanced interlaminar strength, cost savings through near net shape production, and better damage tolerance. However, there are potential disadvantages including lower fiber volume fraction, the existence of resin rich volumes between yarns and degraded in-plane properties (as a result of the non-straight path that the yarns, must accommodate). In many cases the advantages outweigh the disadvantages, and as such, they are extensively used in applications from the sporting goods industry to advanced aerospace vehicles.

Textile composites can pose additional challenges to the experimentalist since they have an additional level of heterogeneity. Not only is there heterogeneity on the fiber-scale, but also on the scale of the textile architecture and the laminate scale (if multiple layers of cloth are used). For many textile forms a yarn may contain 1,000 to 12,000 fibers (or even more), and the repeating unit in the textile architecture may have linear dimensions on the order of 1 cm or more. This can pose significant challenges to the experimentalist, even for seemingly routine tests to determine the elastic properties [115, 116] using strain gages.

It has been assumed that the textile architecture induces repeating spatial variation of strain on the surface coincident with the architecture itself. In order to document the strain distribution associated with the architecture, high spatial and measurement resolution are required. Recent studies using moiré interferometry, have documented the strain field on the surface of composites. These results can be used to guide instrumentation practices for strain gages as well as validate modeling efforts.

Lee et al. [117] used digital phase-shifting grating shearography (essentially moiré interferometry) to experimentally characterize plain-weave, carbon/epoxy composites under tensile loading conditions. They found that the strain varied cyclically and followed the weave architecture. Figure 18 shows contour plots as well as line plots (at two different locations with respect to the architecture) of the tensile strain for four different load levels. The maximum strains were on the order of three times greater than the minimum strain values. In the transverse direction they found that the normal strain is predominantly compressive (Poisson effect), but locally there are regions of tensile strain. These effects are attributed to yarn crimp and bending as a function of axial load.

Fig. 18
figure 18

Tensile strain contour maps for four load levels. Plots along two lines show the variation of strain and correspondence to the textile architecture (Courtesy of J. Lee)

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Shrotriya et al. [118] used moiré interferometry to measure the local time-temperature-dependent deformation of a woven composite used for multilayer circuit boards. They studied both the deformation fields in the plane as well as over the cross-section through a temperature range of 27°C to 70°C. The measurement over the cross-section demonstrates the spatial resolution of the method. Their measurements revealed the influence of the fabric architecture in the deformation field. They noted that the variation in strain was greater when the composite was loaded in the fill direction (versus the warp direction) due to higher crimp angles (the angle that defines the undulation of a yarn as is passes over and under the transverse yarn). They also found that the total deformation increased with temperature and time (reflecting what was previously measured using strain gages) but the shape and distribution remained almost identical for all the loading cases and sample configurations. Figure 19 represents a sample of the fringe patterns taken on the edge of the composite for various times after the load was applied. In the figure, both the horizontal and vertical displacement fields are presented, as well as a schematic of the fabric architecture.

Fig. 19
figure 19

Moiré fringe patterns in the fill direction at 27°C for (a) initial u-field, (b) u-field at 0.1 min., (c) u-field at 1 min., (d) u field at 10 min, (e) composite microstructure, (f) initial v-field, (g) v-field at 0.1 min., (h) v-field at 1 min., (i) v field at 10 min (Courtesy of Shrotriya)

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Damage Initiation

Moiré interferometry was used by Mollenhauer et al. [119] to study damage mechanisms in open-hole tension tests on laminated composites. Figure 20 describes an experimental effort to examine the effects of sub-critical matrix damage (cracks and delaminations) on the strain field in a tensile loaded open-hole composite laminate. In this case, the quasi-isotropic specimen was pre-loaded at several levels to induce various amounts of matrix damage. The figure shows a schematic of the maximum damage extent as well as evidence of the damage influence on the transverse strain component over the whole field of view. After each pre-load, the specimen was examined in a moiré interferometer at a common, lower load. These strain patterns were then compare, as shown in the line graph. As evident in the graph, a strain concentration in the 0° ply is seen at approximately 1 mm from the hole centerline. This strain concentration is coincident with an underlying 45° matrix crack. The significance of this effect can be seen in the failed specimen image where the failure initiation in the 0° ply can be seen to occur at approximately 1 mm from the hole centerline.

Fig. 20
figure 20

Evidence of subcritical damage in the form of cracking and delamination can be seen in the strain field acquired via moiré interferometry (Courtesy of D. Mollenhauer)

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Fracture Mechanics

Inelastic deformation appears to accompany the fracture process, even for brittle materials. It is recognized as permanent deformation in the neighborhood of the crack path, and for nonmetals it is known as the damage wake. Moiré interferometry has been used extensively for experimental fracture mechanics, most notably, crack tip displacement measurement [120–125] and deformation analyses of fracture zone near a crack tip [126–134].

Nishioka et al. [120] used moiré interferometry to investigate interfacial crack-tip behavior of an epoxy/aluminum bimaterial specimen. Kang and Anderson [121] used a combined Moire-Sagnac interferometry method to measure three-dimensional crack tip deformations at room as well as high temperatures. Kang and Lu [122] investigated the displacement fields near the tip of a crack in a bimaterial joint under mixed-mode loading. Savalia and Tippur [125] mapped the debonding evolution between the inclusion-matrix pair in a composite by recording crack opening displacements using moiré interferometry. The representative surface deformations analyzed in the study are shown in Fig. 21.

Fig. 21
figure 21

Selected moiré interferogram of crack-inclusion specimen showing debonding between inclusion and matrix: (a), (b) Before debonding, (c), (d) After debonding (Courtesy of H. Tippur, Auburn University)

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Guo and Kobayashi [126] obtained the crack tip displacement fields from moiré interferometry and used the results interactively to drive a finite element (FE) model of double cantilever beam (DCB). Bastawros and Kim [127] utilized moiré interferometry to document the in-plane components of the Almansi strain tensor at the microstructural length, near a stationary crack-tip in a four-point-bend specimen of polycrystalline aluminum. Tran and Kobayashi [130] used a hybrid experimental (moiré)-numerical (FEA) procedure to analyze the trailing fracture process zone associated with stable crack growth at room and elevated temperature in high density polycrystalline alumina. The crack bridging stress and the dissipated energy in the fracture process zone were determined at room temperature, 600°C, 800°C, 1,000°C and 1,200°C from moiré fringe patterns (Fig. 22). Ma and Kobayashi [131] and Kojaly et al. [129, 132] used the T-epsilon integral to assess stable crack growth and crack linkup in 0.8 mm thick 2024-T3 aluminum tension specimens with multiple site damage (MSD) under monotonic and cyclic loads. Liu et al. [133] utilized high temperature moiré interferometry (HTMI), coupled with SEM/EDAX, to analyze the crack growth behavior of Inconel 718. Ramulu et al. [134] utilized moiré interferometry to characterize the fundamental elastic-plastic stress/strain response of friction stir-welded butt joints in thin-sheet, fine grain Ti-6Al-4 V titanium alloy.

Fig. 22
figure 22

Moiré interferometry fringe patterns of AL23-1998 WL-DCB specimens (Courtesy of A. Kobayashi, University of Washington)

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The deformation fields around cracks obtained by moiré interferometry have been utilized effectively to determine the stress intensity factor [135–137]. Wang et al. [135] used moiré results to determine the interfacial fracture toughness of a flip-chip package subjected to a constant concentrated line load. McKellar et al. [136] compared complete near-crack-tip fields and singular fields associated with the stress intensity factor, where the use of moiré interferometry to measure crack-tip quantities was discussed. Yoneyama et al. [137] proposed a method for evaluating mode I, mode II and mixed-mode stress intensity factors from in-plane moiré displacement fields. This approach was also used to characterize the coupled electro-elastic fracture behavior of piezo-ceramics [138, 139].

Biomechanics of Teeth

In the human tooth, the structural part is a hard enamel shell surrounding a bone-like material called dentin. When restorations are required to treat decay, breakage, etc., the dentin is exposed and effective adhesion of the restoration material to the dentin is required. Moiré interferometry has been used to determine the deformations of teeth, especially for the mechanics of dentin [87, 140–145].

Wang and Weiner [140] first attempted to map the in-plane strain distribution in slices, from human tooth crowns under compression. It was found that the strain inside enamel was much less than in dentin. Kishen and Asundi [141] investigated the adaptation of dentine to temperature variation using digital moiré interferometry. Arola et al. [142] evaluated the fatigue and fracture properties of bovine dentin using the in-plane displacement fields during stable crack growth.

Wood et al. [143] evaluated the effects of changes in humidity on the dimensional changes in dentin disks constrained by enamel and in unconstrained dentin. The moiré fringe patterns in Fig. 23 represent the change in deformation between the moisture states. The results indicated that there were wide variations in strain between the two specimen geometries. Later Kishen and Asundi [145] used a similar approach to investigate the role of free water on the in-plane, mechanical strain response in dentine structure.

Fig. 23
figure 23

Differences in moisture state on human teeth can be seen in the moiré fringe patterns (Courtesy of J. Wood, Clemson University)

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Conclusion

The experimental technique of moiré interferometry is a mature technology and has been utilized in wide variety of applications in solid mechanics and materials. Over the past decade, most of the applications have been in the areas of microelectronics, material characterization, micromechanics, residual stress, composites, facture mechanics and biomechanics. With its high spatial and measurement resolution, it is expected that moiré interferometry will continue to play an important role in the understanding of mechanical phenomena for years to come.