## Abstract

In this paper, we address the behaviour of lacquer coatings similar to that found on the Mazarin Chest, an important Japanese lacquerware artefact currently held by the Victoria and Albert Museum (V&A) in London. The response of Japanese lacquer (*urushi*) to changes in environmental conditions was investigated by examining the deflection of a glass substrate coated with a thin film of *urushi* subjected to changes of humidity. This deflection, measured using phase-shifting interferometry, was then used to determine the two in-plane hygral stress components. Results were compared for two sample conditioning regimes—subjected to intense UV ageing and no ageing—each at a range of relative humidity (RH) steps. The changes in humidity were found to cause rapid stress changes in the lacquers, which then relax over much longer time scales. A simple one-dimensional model of the moisture transport and the stress development is shown to be effective in describing the response of the material to changes in environmental RH.

## 1. Introduction

Lacquers serve two main functions: firstly, protection of an object from the effects of the environment, and secondly as a medium for decoration. In some objects, the lacquer may serve both purposes and an exquisite example of this is the use of natural lacquers, also known as *urushi* in Japan (Kumanotani 1995; Ogawa *et al.* 1998*a*,*b*; Awazu *et al.* 1999; Lu & Yoshida 2003), which have been used on Asian artefacts for centuries. The work described in this paper has been motivated by the need to conserve the lacquer on the artefact known as the Mazarin Chest (Rivers 2003), currently held at the Victoria and Albert Museum. The Mazarin Chest is one of the finest extant pieces of seventeenth-century Japanese *urushi* craftsmanship. Unfortunately, however, environmental conditions have taken their toll on the lacquer surface and conservators are faced with a number of questions regarding its repair.

One of the most common methods used for the conservation of natural lacquer objects is to apply another layer of *urushi* (often diluted) to the damaged surface, which fills holes or micro-cracks caused by damage to the original surface. Unfortunately, although tradition holds that this is generally a successful route, conservators do not know, for sure, whether this is the most effective method, or indeed, whether such a choice can lead to a long-term negative impact.

The logical approach must then be first to establish the properties and response of *urushi* in isolation, before proceeding to consider the interaction between various components of the system as a whole: environment, lacquer layers and object.

Key dependencies of the mechanical properties of *urushi* are the temperature, moisture content and ultraviolet exposure (Obataya *et al.* 2002). For example, Ogawa *et al.* (1998*b*) found that East Asian films had a tendency to toughen by water absorption, leading to an increase in the strain at break and a decrease in the elastic modulus. This behaviour was caused by water serving as a plasticizer. Ogawa *et al.* (1998*b*) also found that the relaxation modulus of the film in a wet condition decreases rapidly with time.

Other studies have demonstrated that the storage modulus decreases and the loss tangent increases with increasing moisture content for both clear and virgin lacquer films (Obataya *et al.* 2002). Moisture can also accumulate at the interfaces between layers generating stresses or can act to reduce the adhesive strength of these interfaces. The result is that delaminations can occur and the lacquer may eventually peel off from the object.

Many other techniques have been used to observe and measure the response of coatings to changes in the environmental conditions (Withers & Bhadeshia 2001; Francis *et al.* 2002), including X-ray diffraction, neutron diffraction, ultrasound, curvature measurements, nano-indentation and Raman–Fourier-transform infrared spectroscopy. A number of authors have used the so-called curvature method to determine residual stresses in thin films and the wide applicability and accuracy of this approach have shown it to be a powerful method (Clyne & Gill 1996; Tsui & Clyne 1997; Schafer *et al.* 1999; Klein 2000; Chen & Wolf 2003; Matejicek & Sampath 2003). In this paper, we present a variant using phase-shifting interferometry (PSI) to determine small strains and stresses in *urushi* thin film (Tien *et al.* 2000).

Interferometry techniques have been used to determine the response of a range of complex materials and are particularly suitable for small magnitude whole-field measurement. Examples of their use include speckle interferometry to measure strain development in foodstuffs (Saleem *et al.* 2003) and composite materials (Maranon *et al.* 2007), moiré interferometry to map strain fields in polymer adhesive joints (Ruiz *et al.* 2009), wavelength scanning interferometry to observe contact forces (Zhou *et al.* 2005) and measure depth-resolved displacement fields (Ruiz *et al.* 2005), to name a few. These methods are powerful for measuring small changes in displacement or strain, and in this paper we will apply a related technique to measure small responses in *urushi* samples when subjected to changes in environmental conditions.

It is clear that the behaviour of *urushi* is highly complex. It is a viscoelastic medium with evidence of thixotropy (Kumanotani 1995), with strong changes in its mechanical behaviour with changes in moisture content and UV exposure. In order to formulate a predictive mechanical model, a comprehensive analysis of the material’s response to changes in environmental conditions is still required. Moreover, as the ultimate cause for the formation of surface micro-cracks is the surface stress, detailed measurements of the dependence of film stress on environmental conditions for both aged and non-aged films are required. In this paper, we present a methodology to achieve this and start by measuring the film stress as a function of changes in relative humidity (RH) using the curvature method (§2). In §3, we describe our experimental technique, based on optical methods, that allows us to determine small deformations of a substrate coated with a thin *urushi* film owing to small changes in the environmental conditions. In §4, we compare the response of an *urushi* layer exposed to a substantial period of elevated temperatures and UV illumination to that of a non-aged layer and finally in §5, we describe a one-dimensional model developed to describe the observed *urushi* behaviour.

## 2. Methodology

### (a) The curvature method

The curvature method, widely used for the determination of residual stresses in thin films, consists of measuring the deflection of a substrate upon which a film is deposited. Under certain conditions, the substrate deflection field *δ*(*x*,*y*) and the average stress *σ* in the film are related by the following approximation, known as Atkinson’s formula (Atkinson 1995):
2.1where *E*_{s}, *v*_{s} and *t*_{s} are Young’s modulus, Poisson’s ratio and thickness of the substrate, respectively. *t*_{f} is the film thickness, assumed to be constant across the substrate, while *r*=(*x*^{2}+*y*^{2})^{1/2} and are the polar coordinates of a point with Cartesian coordinates (*x*,*y*) lying in the plane of the substrate (figure 1). Provided that certain conditions are satisfied, *σ* is independent of the Young modulus and Poisson’s ratio of the film, which makes this method appealing for our purposes as these are currently unknown. The following assumptions (Huang & Zhang 2006) are used to derive equation (2.1):

Strains and rotations are infinitesimally small (i.e. all displacement gradients ≪1).

Film–substrate thickness ratio is

*t*_{f}/*t*_{s}<0.4. When the thickness ratio*t*_{f}/*t*_{s}≪0.1, however, a simpler approximation can be used, known as Stoney’s equation (Stoney 1909).The substrate material is homogeneous, isotropic and linearly elastic, and the film material is isotropic.

The film is in plane stress.

Although the technique was developed with the aim of measuring residual stresses, our concern in this paper is to measure the response of the average stress in *urushi* thin films due to changes in RH. In order to achieve this, several requirements needed to be addressed. First, an appropriate film and substrate system has to be designed so that assumptions 1–4 are all met. Secondly, a sensitive method is required to measure the small deflections expected of the substrate (§2*b*). Finally, a chamber is needed for full control over the system environment (§3*c*).

### (b) Non-contact deflection measurement using phase shifting interferometry

PSI is a well-known optical technique that allows full-field, non-contact measurements of sub-micrometer deformations without sign ambiguities (see for instance Huntley (2001)) making PSI an ideal candidate for measuring small substrate deflections. Light from a laser is split into reference and object beams, which upon reflection from a reference mirror and the object surface are recombined on a detector array through an imaging system to form a fringe pattern described by the following intensity distribution (Huntley 2001):
2.2*I*_{o}(*x*,*y*), *I*_{M}(*x*,*y*) and *ϕ*(*x*,*y*) are three unknown two-dimensional distributions referred to as the background intensity, the intensity modulation and the phase difference between the interfering beams, respectively, and *x* and *y* are spatial coordinates. In order to evaluate the phase *ϕ*(*x*,*y*), at least three independent measurements of the intensity *I*(*x*,*y*) are required. In order to measure object deformations, two-phase distributions *ϕ*_{r}(*x*,*y*) (phase of the reference state) and *ϕ*_{d}(*x*,*y*) (phase of the deformed state) are required to evaluate the phase-change distribution. We use a variation of the well-known four-frame algorithm, which is based on four interferograms with *π*/2 phase shifts between the interfering beams for the reference state and four for the deformed state. These are then combined to extract the phase change **Δ***ϕ*_{w}(*x*,*y*) wrapped in the range −*π* to *π*, enabling low-pass filtering to reduce phase noise at the same time (equation 2.52 in Huntley (2001)). Finally, the continuous phase-change distribution **Δ***ϕ*(*x*,*y*) is obtained by adding an appropriate integer multiple of 2*π* at each point in the wrapped phase distribution (Ochoa & Huntley 1998; Huntley 2001). In the case when the interferometer has pure out-of-plane sensitivity, i.e. parallel to the observation direction, the relationship between the measured unwrapped phase change **Δ***ϕ*(*x*,*y*) and the displacement (or deflection) distribution is given by (Rastogi 2001)
2.3where *λ* is the laser wavelength. Equation (2.3) assumes that the object and the interferometer are immersed in a medium of unit refractive index.

### (c) Evaluation of film stress from displacement field measurements

Ideally, a film and substrate system such as the one described in §2*a* will respond to a uniform stress field in the film by deforming with axial symmetry so that a map of deflection perpendicular to the *x*−*y* plane has circular contour lines. However, slight heterogeneities in the stress field lead to elliptical contours, as shown in figure 2. The principal axes of the elliptical contour lines correspond to the directions of the principal in-plane stress components in the film plane.

Ignoring the linear and constant terms, the deflection distribution *δ*(*x*,*y*) can conveniently be approximated by a second-order surface as follows:
2.4where the quadratic coefficients *a*, *b* and *c* reflect the curvature characteristics of a paraboloid.

Once the orientation of the ellipse is determined, the problem is reduced to the determination of *δ* along the minor and major axes of the ellipse, which subtend angles *θ*_{m} and *θ*_{M}=*θ*_{m}+(*π*/2) to the *x*-axis, respectively. Substitution of equation (2.4) into equation (2.1) for *θ*_{m} and *θ*_{M} finally leads to the film principal stresses
2.5Owing to the fact that a higher film stress would be responsible for the curvature of the substrate on the plane that contains the minor axis of the ellipse and the *z*-axis, *σ*_{1} and *σ*_{2} have been defined here following the usual convention in which *σ*_{1}>*σ*_{2}.

If *δ*(*r*,*θ*) is available as a full-field measured deflection distribution, then coefficients *a*, *b* and *c* can be found by least-squares fitting of an elliptical paraboloid to the measured distribution. The fitted paraboloid is then used to find the orientation *θ*_{m} of the minor axis by minimizing the radius at a contour of constant deflection. The orientation *θ*_{M} of the major axis is found from *θ*_{M}=*θ*_{m}+(*π*/2) and finally equation (2.5) leads to the film’s principal stresses.

## 3. Experiment

### (a) Optical set-up

Figure 3 shows a schematic of the interferometer used to measure changes in the curvature of film–substrate owing to variations of RH. A He–Ne laser provides a vertically polarized 30 mW beam with wavelength *λ*=632.8 nm. Half-wave plate (HWP) and polarizing beam splitter (PBS) divide the incoming beam into two beams with orthogonal polarizations, the intensities of which can be easily adjusted by rotating the HWP. The reference beam (RB) goes through a pair of glass wedges, one of which is fixed and the other is moved across the beam with an open-loop piezoelectric lead zirconate titanate (PZT) transducer to increase the optical path and introduce controlled phase steps. The procedure described by Ochoa (Ochoa & Huntley 1998) was followed to calibrate the PZT actuator.

The beams are then launched into single-mode polarization preserving optical fibres which deliver them with the same polarization to the recombination head of the interferometer. The object beam (OB) is transmitted through a non-polarizing beam splitter (NPBS) and propagates towards the film–substrate sample, at which point it is reflected back from the specularly reflecting bottom surface of the substrate. On its way back, the OB is recombined with the RB so that the optical path difference remains within the coherence length of the laser. A CMOS camera C (HCC-1000 Vosskühler, 8 bits, 1024×1024 pixels) records the interference fringe patterns that encode the shape of the substrate relative to the reference wave-front. The purpose of lens L_{1} is to illuminate the sample with a collimated beam, and thus the distance from the tip of the object optical fibre is equal to the focal length of L_{1}. L_{2} collimates the beam launched by the reference fibre and lens L_{3} focuses it at the aperture stop plane of imaging lens L_{4}.

### (b) Preparation of non-aged and UV-aged *urushi* films

The lacquer used in this study was *kijiro* *urushi* from Watanabe Syoten Co., Japan. The first step was to filter in the traditional Japanese way by squeezing the *urushi* through Rayon paper to remove any large impurities. As *urushi* is an unstable water-in-oil type emulsion, the water-soluble polysaccharides in the raw *urushi* can often aggregate during curing (Obataya *et al.* 2002). In order to avoid this, raw *urushi* was mixed and homogenized for about 3 min requiring gentle mixing to prevent the formation of air bubbles. Thin films of *urushi* on circular glass substrates (BK-7, 22 mm diameter and 190±5 μm thickness) with a thickness of around 20 μm were produced by spin coating at room temperature for 90 s at 3000 r.p.m. The back surface of the glass substrate was mirror-coated in a vacuum deposition chamber to increase its reflectivity and thus obtain high-visibility interference fringes. *Urushi* cures only in the presence of high RH. For films of around 20 μm, it takes at least 3 days to ensure that they are fully cured at a humidity level of 75±2% RH used here. The film thickness was measured by focusing a microscope (BX-60 Olympus with 50×objective) on the glass–air and the *urushi*–air interfaces and measuring the distance required to refocus. We obtained films with a thickness of 21±2 μm, which resulted in a film–substrate thickness ratio of about 0.11.

The process just described was used to produce ‘non-aged’ *urushi* films, without subjecting them to any form of subsequent degradation. One subset of the non-aged *urushi* films was then exposed to UV radiation in order to simulate UV ageing and observe its effect on the stress response to changes in RH. The film was exposed to 340 nm, 0.7 Wm^{−2} UV radiation for 400 h in a Q-Sun environmental test chamber equipped with a xenon arc source.

### (c) Environmental chamber

Triggering the response of *urushi* films to changes in RH and measuring it with the interferometer requires a fine control over the environmental conditions. An environmental chamber (EC) was built to control RH and temperature to within ±1% RH and ±1^{°}C, respectively.

The film–substrate test sample sat horizontally on a recessed holder. A steering mirror was used in the OB as shown in the insert in figure 3 to direct it vertically so that the bottom surface of the substrate acted as a specularly reflecting surface, the shape of which is measured with the interferometer described in §2*b*.

### (d) Gravimetric measurements

Diffusion of water in *urushi* films was studied by the gravimetric method in which the changes in weight of all the samples were monitored as a function of time by weighing them on an electronic balance (HA180 A&D Instruments Ltd) to a precision of ±0.1 mg. During the experiments, the electronic balance was kept within a second EC to ensure a controlled humidity and temperature.

### (e) Desorption and sorption measurements

The diffusion of the moisture through the lacquer was characterized using desorption and sorption experiments involving four-step changes in RH, for both non-aged and aged *urushi* films. The samples were prepared to a uniform moisture content by storing them in the curing chamber at 75 per cent RH. They were then suddenly exposed to 30 per cent RH for 20 h. After that the environment was changed to 40 per cent, 50 per cent and 60 per cent RH for 16, 20 and 20 h, respectively, and the samples’ masses were observed every 30 min. The samples were dried at 100^{°}C for 27 h to remove any excess moisture and then weighed.

During exposure to each RH level, the moisture content of the samples was measured gravimetrically and calculated using
3.1where *M*_{t} is the moisture content (weight per cent) at time *t*, *m*_{t} is the mass of the specimen at time *t* and *m*_{dry} is the mass of the dry specimen. Having determined the moisture uptake for both aged and non-aged *urushi* layers, we used a diffusive model of mass transport to estimate the diffusion coefficient.

The mass transport of moisture is given by the diffusion equation (Crank 1975; Shirrell 1978; Shewmon 1990; LaPlante & Lee-Sullivan 2005)
3.2where *C*(*z*,*t*) is the moisture concentration, *t* is time, *z* is the spatial coordinate in the through-thickness direction and *D* is the diffusion coefficient. The air–*urushi* interface is chosen to lie at *z*=0 and the *urushi*–glass interface at *z*=*t*_{f}. However, it is convenient to solve equation (3.2) for a film of twice the true thickness *t*_{f} (i.e. over the region 0≤*z*≤2*t*_{f}) as the symmetry of the solution for *C* then ensures zero concentration gradient and hence zero moisture flux at the *urushi*–glass interface.

#### (i) Initial conditions

In the case where the initial concentrations of the penetrant are uniform throughout the film, we take the initial conditions to be
where *C*_{°} is the initial concentration and *t*_{f} is the thickness of the film.

#### (ii) Boundary conditions

If the film is exposed to different concentrations of penetrant from the one to which it was originally exposed, then the boundary conditions are
where *C*_{s} is the surface moisture concentration corresponding to the environmental RH.

The solution to equation (3.2) (Crank 1975) is well known and can be found in series form
3.3where is the moisture content at equilibrium and *A* is the exposed surface area of the film.

### (f) Experimental errors

An error propagation analysis was performed to estimate the error in the calculated stress. A maximum applied RH change from 75 to 30 per cent to non-aged *urushi* resulted in a maximum stress of 11.3±1.4 MPa (§4) using the experimental parameters *t*_{s}=190±5 μm, *t*_{f}=21±2 μm, *δ*=14.32±0.17 μm and *r*=8±0.018 mm. The film thickness has the greatest contribution to the error of the estimated average film stress while the displacement and the distance, from the centre to where the displacement is measured, have the lowest contributions. The overall uncertainty of approximately 1.4 MPa for the state described suggests that we should expect errors of the order of less than 12 per cent in the calculated absolute stress values.

## 4. Response of *urushi* to changes in moisture

The diffusion coefficient was obtained using the solution given in equation (3.3), fitted to the experimental data using regression techniques. The resultant values for the diffusion coefficient, the initial and asymptotic moisture contents and the coefficients of determination (*R*^{2}) for non-aged and aged *urushi*, are given in table 1. Typical fits are shown in figures 4 and 5. Intermediate moisture contents at equilibrium can be obtained by interpolating the asymptotic moisture content using a quadratic fit, where *R*^{2} is about 0.999, and solving for the required values at 30, 36 and 42 per cent RH (figure 6). Thus, using the initial and asymptotic moisture contents, and the fitted *D* value, the depth-averaged moisture content can be calculated as a function of time using equation (3.3) at 30, 36 and 42 per cent RH for non-aged and aged films of thickness 21 μm. This will then also allow the depth-averaged stress to be calculated in §5*a*, with further extension to the prediction of stresses within multiple layers to be discussed in §5*b*.

A contour representation of a typical displacement distribution (figure 2) shows the substrate deflection field where the orientation of the minor axis, *θ*_{m}, is about (3/4)*π* and the orientation of the major axis, *θ*_{M}, is about (5/4)*π*. For this figure, the depth-averaged film stresses along the minor and the major axes, *σ*_{1} and *σ*_{2}, are 9.14 and 7.51 MPa, respectively.

The stress response of three non-aged and three UV-aged *urushi* films to changes in RH were measured in this work. All the samples were prepared as described in §3*b*. Each of them was used for measuring the response to a particular change in RH, rather than using the same sample to measure different RH changes, so as to avoid history-dependent effects. Initially, all samples were kept at the RH for curing, RH_{C}=75±2%, for three weeks to ensure equilibrium. Each sample was then exposed to a step reduction in RH, to one of three different low RH levels, denoted as RH_{1}. Under each low RH, the film stress was observed over 66 h, a time that corresponds to 95 per cent of an approach to the asymptote. The measurements were carried out every 5 min until the stress reached a maximum value and then every 1 h. During all measurements the temperature was held constant at 23^{°}C.

Figure 7 shows the development of average film stress in three different non-aged and aged *urushi* films during exposure to the low RH levels, as a function of time. The general behaviour of the *urushi* film following a reduction in humidity from 75 per cent is a positive deflection of the substrate (cusp towards the camera), which corresponds to a tensile in-plane stress developing in the film. This stress peaks within about 2 h then relaxes over a longer time scale with a slight reduction in the magnitude of the stress. We can attribute different physical mechanisms to the stages of deformation for the material when subjected to a reduction in RH. Shortly after the humidity is changed, the desorption of water will lead to volume shrinkage and as a result of the adhesion to the glass substrate a tensile stress develops in the film, balanced by a compressive in-plane stress in the substrate. The in-plane stress soon peaks, however, and is followed by relaxation of the material. It is likely that the viscoelastic properties of the *urushi* cause this time-dependent stress relaxation since the humidity, and, therefore, hygral strain is maintained at a constant value once hygrothermal equilibrium is reached. The peak stress values and times for non-aged and aged *urushi* samples, which were subjected to a reduction in RH levels, are shown in table 2 and are plotted in figure 8 as a function of the difference between storage (75%) and the different target RH levels. This shows clearly that the absolute value of the in-plane stress scales with the size of the RH change.

The form of the behaviour observed following a humidity change is broadly consistent regardless of whether the material has been aged or not, but we do observe significant differences in the absolute values. In general, for large reductions in RH, the aged films exhibit a smaller in-plane stress than the non-aged materials. For example, we observe a approximately 30 per cent reduction in the peak tensile stress in figure 7, which is replicated for other changes in RH, and applies to the asymptotic stress as well.

## 5. A one-dimensional model of stress development in *urushi* layers

Having observed the response of the material to changes in RH, a one-dimensional model is now proposed which will be used to characterize and predict *urushi* behaviour. We will show how this model can describe the observed behaviour of the *urushi* thin layers and then go on to demonstrate the model’s potential for applications in this area.

### (a) Hygral stresses induced by absorption of moisture

The experimental observations of the stress response over time indicate that the system responds to both changes in moisture content and relaxation of the stresses in the material, with the moisture changes dominating initially, followed by a period in which the relaxation has a stronger effect. Under these conditions, we can consider an isotropic plate in plane stress subject to the biaxial mechanical stresses *σ*_{1} and *σ*_{2}, moisture change **Δ***C* and temperature change **Δ***T*. In this case, the strain components *ε*_{1} and *ε*_{2} are given by
5.1where *α* is the thermal expansion coefficient, *β* is the hygroscopic expansion coefficient, *ν* is Poisson’s ratio and *E* is Young’s modulus. For the case where *σ*_{1}, *σ*_{2} and **Δ***T* are all zero, we are left therefore with the hygral strains assuming that the film is isotropic. The film is however unable to expand owing to the underlying substrate, giving rise to hygral stresses which using force superposition can be given by
5.2To describe the stress relaxation over time, we employ a three-element viscoelastic material model given by the equation (Haddad 1995)
5.3where *τ*_{°} is the time constant for the model, *ε*_{°} is the magnitude of the imposed strain step, and and *E*_{°} are, respectively, the relaxed and instantaneous moduli (Haddad 1995). The effective relaxation function, *R*, is therefore
5.4The stress for a general strain history, *σ*(*t*), is
5.5which for the model considered here, with *σ*(*t*)=0 for *t*<0, can be written
5.6where
5.7Thus, the material response to a change in moisture content is characterized by the three parameters *k*_{1}, *k*_{2} and *τ*_{°}, which are themselves combinations of the usual material properties.

The values of *k*_{1}, *k*_{2} and *τ*_{°} for each combination of RH and ageing can be determined from the experimental data shown in figure 7 by fitting equation (5.6) to the data. The term d*C*/d*t*′ is proportional to the strain rate and using the solution given in equation (3.3), we can determine the depth-averaged moisture content as a function of time and calculate the rate.

The fitting of equation (5.6) is reasonably good over the time ranges interrogated for all three curves (figure 7) and rheological parameters *k*_{1}, *k*_{2} and *τ*_{°} obtained for non-aged and aged *urushi* are shown in tables 3 and 4. The parameters show no obvious dependency on RH but there does appear to be a systematic upward shift in relaxation time *τ*_{°} as the humidity level rises for the non-aged material. For the aged material, the picture is similar, although the data are somewhat confused for the shift from 75 to 42 per cent RH.

### (b) A one-dimensional model for multiple layers of *urushi* on a substrate

The agreement between the one-dimensional model and the experimental observation suggests that the important behaviour is being captured by the model. In practice, *urushi* is most often found in multiple layers. A natural extension to the model, therefore, is to incorporate further layers of *urushi*. To demonstrate the possibility, we consider a domain consisting of two layers of *urushi*, constrained in the lateral direction by the presence of an infinitely stiff and impermeable substrate as the lower most boundary and a completely permeable interfacial layer. The moisture content at the upper most boundary is fixed by the RH in the environment.

The problem is solved sequentially. First, we solve the diffusion equation using a finite-difference approximation to find the moisture content distribution within each layer. A time step of 0.01 min was employed throughout the analysis, and the number of spatial nodes varied until we obtained a mesh independent solution. The depth-averaged moisture content in each layer was then calculated from the time-dependent moisture profile, and the moisture rate determined. The initial rate was obtained by examining the short-time behaviour at successively shorter time steps until the initial rate was independent of the time step. This rate was then taken as the rate of the depth-averaged moisture content at *t*=0. The boundary conditions were a fixed moisture content at the upper surface specified by the relationship in figure 6, a zero-flux condition at the base representing an impenetrable barrier and we assume that the flux at the layer–layer interface is controlled by a diffusion coefficient, which is an average of the two layers’ diffusion coefficients. The model was validated by comparing against the numerical solution expected for a single layer of the same depth, and the analytical solution of a single layer. The rate of the depth-averaged moisture content for each layer was then inserted into equation (5.6) and the stress calculated using the material properties *k*_{1}, *k*_{2} and *τ*_{°} determined previously in §5*a*.

This model was used to investigate the development of stress as a function of upper layer depth when placed upon a much thicker support layer. The bottom layer was set to be 500 μm, and represented an aged layer, while the thickness of the upper layer was set to be 5, 10, 20 and 50 μm. We then envisage a scenario where there was a step change in RH (25% RH) resulting in a change of moisture content of −1.31 per cent and then used our model to determine the stress evolution. Figure 9 shows the depth-averaged stresses in the upper and bottom layers, respectively. We see that the stress developed in the upper layer is significantly larger than that in the lower layer, but that the form of the stress distribution is similar. This simple model suggests significant complexity in the stress development in lacquers where conservation may have occurred and the layers are of significantly different age and depth. In both cases, we observe that reducing the depth of the upper layer will result in an increase in both the maximum and asymptotic stresses, which can be attributable to the quick uptake of moisture and the rapid passing of moisture through the upper layer to the lower, when using thin upper layers. This model has the benefit of simplicity and may in the future be modified to include time-varying RH, multiple layers and the effect of changes in the substrate. It is hoped that it will be effective in guiding the type of layers that should be employed to reduce the magnitude of stresses that a conserved *urushi* lacquer might suffer in environments of rapidly changing moisture content.

## 6. Conclusions

The stress response of cured non-aged and aged *urushi* films to different humidity levels has been measured using a phase-shifting interferometer. This method has been shown to be able to resolve displacements in a bilayer to within 1 per cent of the best second-order deflection approximation. We compared the behaviour of aged and non-aged *urushi* films when subjected to changes in the environmental RH. We observed similar modes of behaviour, with a strong indication of time-dependent behaviour and a coupling of the diffusion and relaxation of the material, but we also observed that the in-plane stresses during desorption were higher for the non-aged *urushi* films. The results indicate that the material properties are likely to be strongly affected by the moisture ingress during variation of the RH, and this depends on the amount of ageing the material has been subjected to. A one-dimensional model has been developed and tested against the experimental observation, showing reasonable agreement. We have demonstrated the potential of our model by investigating the stress response in layers of *urushi* of different ages. Further work is required to identify the material constitutive relationships in order to predict and describe *urushi* material behaviour under varying environmental conditions.

## Acknowledgements

The authors would like to acknowledge the support of the Victoria and Albert Museum (V&A) and the Toshiba International Foundation (TIFO). A.E. would like to thank Loughborough University for scholarship contributions. Software developments by Russell Coggrave are gratefully acknowledged.

- Received August 4, 2010.
- Accepted October 25, 2010.

- This journal is © 2010 The Royal Society