1 Introduction

Soft materials are used in important components in a wide range of products such as flexible electronics, surgical robots, and vibration absorbers. The nature of contact between a soft material and a rigid object is very important for many of these engineering applications. However, these materials present challenges to both the prediction and measurement of strain fields due to the large deformation that they exhibit under relatively modest levels of applied loads.

A rigid indenter pushed into an elastic body is representative of a number of engineering applications and has attracted attention of many researchers. The classic paper of Boussinesq^[1] first considered the distribution of stress within an elastic half space indented by a rigid punch. Based on this solution, Sneddon^[2] derived some simple expressions for the distribution of pressure for several punch shapes. Subsequently, there have been several studies, e.g., Refs.^[3] and ^[4], concerned with the subsurface deformation of material produced by the indentation of wedges. Korsunsky^[5] showed that small variations in punch geometry have significant influence on the contact strains. Gao and Gao^[6], Chen and Gao^[7], and Gao and Qian^[8] explored the behavior of soft materials indented by rigid indenters from a theoretical perspective and suggested a sector division method, in which the strain field in the soft material block was partitioned into shrinking (SH) and expanding (EX) sectors that can be modelled separately.

It is worth mentioning that, although the aforementioned theoretical models can provide solutions to some specific engineering problems, the validity of these models tends to be limited, because most of the following assumptions are usually adopted: small elastic strains; the absence of shear tractions, i.e., the contact is frictionless; the contact zones are much smaller than the characteristic dimension of the bodies; similar elastic properties for the contacting bodies; and a small wedge angle. There is still a sparsity of experimental data describing the full-field deformation of soft elastic material during indentation by a rigid wedge, and an experimental observation of the sector division concept is also lacking. Thus, the principal aim of this work is to provide more detailed data from the experiments for use in evaluating and developing theoretical models. This is achieved by extending the prior work that used a single wedge geometry^[9]. Specifically, the measured data fields are used to confirm the existence and evolution of the sectors identified by Gao and Gao^[6] and to define the limits of applicability of the small deformation theory proposed by Ciavarella et al.^[10].

2 Experimental apparatus and procedure

The experimental arrangement used in this study is the same as that used by Tan et al.^[9], shown schematically in Fig. 1(a). The stereoscopic digital image correlation (DIC) is used for this work since traditional methods such as strain gauges are not suitable for measuring large deformation over a large field of view^[11]. The DIC has many advantages such as being non-contact, high precision, real-time, and having a wide measurement range and a simple optical setup^[12-13]. In brief, a rectangular block of silicone rubber with dimensions 60mm$\times 60$mm$\times 30$mm is indented by an aluminium wedge of thickness 30mm with a tip radius of 1.68mm. Four wedges are used with the slope angles of 5$^{\circ}$, 15$^{\circ}$, 45$^{\circ}$, and 73.5$^{\circ}$ as shown in Figs. 1(b)-1(e). One of the square surfaces of the rubber block is sprayed with an extremely thin layer of white paint (Matt Super White 1 107, Plasti-kote, U. K.) on top of which was sprayed an even finer dusting of black paint (Matt Super Black 1 102, Plasti-kote, U. K.) to form a speckle pattern, which is used to perform stereoscopic DIC. A commercial DIC system (Q-400, Dantec Dynamics GmbH, Germany), is used which has two cameras (Fire Wire, 1/8'', 1624$\times $1234 pixels) with 50mm focal length lenses resulting in a 35 pixels/mm magnification. The wedge is driven into the block at a constant speed of 1 mm/min, while the specimen is illuminated with two green LED lights and pairs of images are captured at 30 seconds intervals.

The elastic modulus of the wedge and block are, according to the manufacturer's data, linebreak 73 GPa and 2 MPa, respectively, providing a large elastic dissimilarity and that the contact surfaces are dry, i.e., there is no lubricant introduced. Difficulties often occur when the DIC method is applied to large deformation measurements because of serious de-correlation between the reference image and the deformed image. Hence, a mixed Eulerian-Lagrangian approach is used in the DIC, in which the reference images are updated when the level of deformation is sufficient to cause a loss of correlation. This approach is computationally more efficient than updating the reference image at each increment and reduces the potential errors introduced during updating. The sub-image or facet size for the correlation process is 25 pixels ($\equiv $ 0.7mm), and the facets are overlapped by 22 pixels to give displacement vectors at a pitch of 3 pixels or 0.084mm. A typical example of the surface displacement fields for the 15$^{\circ}$ wedge with a nominal indentation depth of 3.3mm ($\equiv $19.1 N) is shown in Fig. 2 with rigid body motions removed using the algorithm provided in the system software. The block always returns to its undeformed shape after indentation, i.e., all of the loading is elastic, and there is no evidence of any adhesion between the indenter and the block.

The uncertainty in the DIC measurements using the set-up described above was previously assessed^[9] using a protocol developed specifically for the purpose^[14] and found to be 31.7 $\mu $m in-plane and 0.65 $\mu $m out-of-plane.

3 Experimental results 3.1 Displacement fields

The measured surface displacement fields are compared with the predicted displacement fields from the solution due to Ciavarella et al.^[10]. According to this solution for the half plane, the interior stress field is given by the following relations:

where $z=x+{\rm i}y$ is the complex form of the coordinates $x$ and $y$ which are defined as shown in Fig. 1 with the origin at the centre of contact. The corresponding Muskhelishvili's potential is

where $p(t)$ is the pressure distributed on the contact boundary $\Gamma$.

In this work, the theoretical stress state in the region of interest is obtained by a numerical solution of (1) and (2). Following the assumptions made by Ciavarella et al.^[10], there are two viable options for computing the interior displacement fields. They can be either deduced directly from the Muskhelishvili potential, or integrated from the corresponding strains that are converted from the stress state using Hooke's law^[15]. It is found that the computation method has little effect on the final results, and the second option is used because it is computationally more convenient. A quantitative comparison is made between the displacement fields measured in the experiments and those obtained from the theory. In this process, each field of displacement is treated as an image and decomposed using Tchebichef kernels^[16] by a specially-written MATLAB program. This process yields a set of mathematical moments, which are a unique representation of the displacement field. The accuracy of the representation is checked by reconstructing the data fields from the Tchebichef moments. A 99% correlation between the original and reconstructed data fields is obtained by the first fifty Tchebichef moments, and this is believed to be sufficient^[16-17]. This decomposition process is invariant to translation, scale, and rotation, while at the same time, it decreases the dimensionality of the data and hence is a powerful tool for making quantitative comparisons^[17], especially when the data fields do not share a coordinate system or data pitch, as is the case here. The results from four indentation depths are analyzed, which correspond to the compressive forces of 3.7 N, 11.4 N, 19.1 N, and 26.8 N, respectively, and the measured and predicted results are shown in Figs. 3 and 4.

It is inevitable that uncertainty is present in the measured data, and there is a further contribution from the decomposition process. These uncertainties are represented in Figs. 3 and 4 by a zone bounded by two parallel lines defined by

where $S_{\rm E}$ and $S_{\rm M}$ are the sets of moments representing the displacements fields from the experiment and model, respectively, and $u(S_{\rm E})$ is the uncertainty present in the experimental data after decomposition and is the root mean square value of the minimum measurement uncertainty^[14] and the decomposition uncertainty. The values of these uncertainties are found to be 28.76 $\mu $m and 8.11 $\mu $m, respectively, using procedures described previously by Patterson et al.^[14] and Tan et al.^[9], respectively. For perfect agreement between the measured and predicted results, all of the data points in Figs. 3 and 4 lie on the line $S_{\rm M}=S_{\rm E}$, but agreement can be considered acceptable and the model valid when the data points lie within the zone described by (3)^[16]. This occurs in Fig. 3 at indentation depths between 0 mm and 2.2 mm for $u$ and $v$ displacements. In order to observe the trends more clearly, linear regression lines are fitted to each set of data and should have a gradient close to one when the model is valid. In this case, the theory is becoming a poor representation when the depth of indentation reaches 3.3 mm, because the regression line for the $u$ displacement just falls outside of the uncertainty band at the extremes of the graph. The theory is not invalid at this depth of indentation because none of the data falls beyond the uncertainty zone. However, the theory can be considered invalid when the indentation depth increases to approximately 4.3 mm, since the data points for both displacement fields lie outside the uncertainty band. Similarly, this analysis method is also applied to the rest of the wedges, and the results are shown in Fig. 4.

3.2 Strain fields

The strain field in the experiments can be derived from the spatial derivatives of the measured displacements. Rubber and rubber-like soft materials, which can experience very large elastic deformation before failure, are known as hyper-elastic materials. They experience large displacements and strains when subject to loading by a rigid wedge, and thus present a very complicated mechanical behavior that cannot be modelled using the linear elastic theory. Instead, it is more appropriate to use the finite deformation elasticity theory, in which the strain fields in a polar coordinate system are described by^[18]

where $(r, \theta )$ are polar coordinates in the reference configuration, as illustrated in Fig. 1, and $u_{r}$ and $u_{\theta }$ are the radial and circumferential displacements, respectively. Since the experimental displacement field is obtained in a Cartesian coordinate system, it is necessary to use the following relationships to evaluate the partial derivatives of $u_{r}$ and $u_{\theta }$:

where $(x, y)$ are the Cartesian coordinates shown in Fig. 1, and $u$ are $v$ are the horizontal and vertical displacements, respectively. The experimental radial strain ${{\varepsilon }_{r}}$, the circumferential strain ${{\varepsilon }_{\theta }}$, and the shear strain ${{\varepsilon }_{r\theta }}$ in the polar coordinate system can be obtained by substituting (4) into (3). Typical results are shown in Fig. 5 for the 15$^{\circ}$ wedge with an applied indentation depth of 3.3 mm ($\equiv $ 19.1N).

4 Discussion 4.1 Displacement fields

The displacement data from the experiments and from the theory due to Ciavarella et al.^[10] are compared for the 15$^{\circ}$, 45$^{\circ}$, and 73.5$^{\circ}$ wedges in Fig. 4. For the 15$^{\circ}$ wedge, all of the data for $u$ and $v$ displacements lie within the uncertainty band for depths of indentation from 0 mm to approximately 3.7 mm, which implies that the theory can be taken as confirmed for these levels of indentation. However, the theory becomes invalid at some depth of indentation between 3.7 mm and 4.7 mm, since for an indentation of 4.7 mm, some of the data points lie outside the uncertainty band.

The regression line gradient progressively deviates from unity with the increasing depth of indentation, implying that the theory is an increasingly inaccurate representation of the experiment. For the 45$^{\circ}$ wedge, the theory is apparently less good than that for 15$^{\circ}$ wedge since it becomes invalid at an indentation depth of greater than 2.8 mm but less than 4.54 mm with the scatter of the data points increasing and a growing number lying outside the zone defined by (3). Not surprisingly, the theory breaks down even earlier for the 73.5$^{\circ}$ wedge with some data outside of the acceptable zone at an indentation depth of 3.2 mm and the regression line gradient close to zero, signifying that the theory is no longer applicable to this specific case.

The tendency in Figs. 3 and 4 is clear, i.e., the sharper the wedge, the smaller the indentation at which the theory fails. It should be mentioned that, although the geometry used in the experiments and for the theoretical solution is the same, the theory describes an idealized case that includes the following assumptions: the half-plane hypothesis is justifiable, zero interfacial friction, elastically similar contacting bodies, very small external wedge angle, and sin $\theta \approx \theta $. Thus, the results are not unexpected since more assumptions in the theory are contravened when the depth of indentation is greater. However, it is noteworthy that the theory breaks down at smaller values of indentation for the 5$^{\circ}$ than the 15$^{\circ}$ wedge. In order to find out a reasonable explanation for this, a theoretical solution for wedge-shaped blunt punch^[19] is evaluated and compared using the same procedure. The interior stress distribution on the half plane can be expressed, using the wedge-shaped blunt punch theory as

The displacement fields are found using the same method as mentioned in Section 3.1 and subsequently decomposed as described above to yield moments which are compared with those obtained from the experiments as shown in Fig. 6 for the 5$^{\circ}$ wedge. The solution remains valid for indentation depths from 0 mm to 3.3 mm but is not valid at an indentation depth of 4.3 mm, although only one point is outside of the uncertainty zone, which is an apparent improvement on the solution due to Ciavarella et al.^[10].

It appears that by increasing the indentation depth, there exists a critical value of indentation at which these theories are no longer applicable to this particular problem, probably because both are founded on linear elasticity. This critical value can be identified when the regression lines in Figs. 3, 4, and 6 no longer fall in the zone defined by (3) for values of $S_{\rm M}$ and $S_{\rm E}$ in the range of the calculated moments, i.e., $-$100 $\mu $m to 200 $\mu $m, which gives rise to the concept of limiting gradients that fit within the uncertainty zone, as shown in Fig. 7 for the 5$^{\circ}$ wedge. Figure 7 also shows the regression line gradients for the horizontal and vertical displacements obtained from the graphs in Fig. 3 as a function of the critical value of applied load. Similar plots are produced for each wedge angle examined, and the critical values are shown in Fig. 8 on the load-indentation curves for each experiment. The critical value for the theory due to Ciavarella et al.^[10] increases linearly from an indentation depth of about\linebreak 1.5 mm for the 75$^{\circ}$ wedge through about 2.75 mm for the 45$^{\circ}$ wedge to about 4.3 mm for the 15$^{\circ}$ wedge, and then drops to about 3.1 mm for the 5$^{\circ}$ wedge. The critical indentation depth for the blunt punch theory applied to the 5$^{\circ}$ wedge is also shown.

Ciavarella et al.^[10] assumed that the blunt wedge had a rounded tip, whereas the curvature at the wedge tip was ignored in the blunt punch theory. The difference in geometry causes a difference in the predicted distribution of contact pressure which in turn affects the displacement field. In the experiment, all of the indenters are manufactured with the same tip radius R=1.68 mm. Therefore, the half-width of the tip becomes smaller as the angle of the wedge decreases. In this case, the half-width of the tip is 0.145 mm for the 5$^{\circ}$ wedge, which is negligible when compared with the contact width at higher indentation depths and probably accounts for the better fit of the blunt punch theory than Ciavarella's solution to the measured data.

4.2 Strain fields

Although Gao and Gao^[6], Chen and Gao^[7], and Gao and Qian^[8] divided the strain field created in a rubber block into EX and SH sectors for which the strain fields can be found separately, there has been no experimental verification of this sector division methodology. The circumferential strain field observed in Fig. 5 appears to be composed of two axially symmetry sectors, i.e., an EX sector directly below the contact and an SH sector on either side of the contact. The transition between these sectors can be taken as the null or zero-valued contours of the circumferential (or radial) strain. These boundaries for five loading steps are plotted in Fig. 9 for each indenter. The opening angle between the boundaries is estimated using a regression line fitted to each boundary and included as a label in Fig. 10. By comparing the plots in Fig. 9, it is obvious that the expanding sector becomes larger when the wedge-shaped indenter is sharper, and conversely the expanding sector becomes smaller and narrower with the increasing load. For example, the opening angle gradually decreases from 159$^{\circ}$ to 100$^{\circ}$ for the 5$^{\circ}$ wedge as the applied load increases from 3.7 N to 34.5 N. These trends are summarised in Fig. 10 where the opening angle of the boundary between the EX and SH sectors is plotted as a function of applied load for each wedge angle. The trends appear to be asymptotic with load implying that the shape of the sectors will stabilise at high loads and remain constant until failure in the material of the block.

It is interesting to observe that EX, the expanding sector becomes smaller by increasing the load, and it is difficult to interpret this trend with any certainty. However, it is reasonable to speculate that when the wedge angle is very shallow, i.e., 5 to 15 degrees, then we can consider a small element $A$ located in the EX sector and close to both the contact surface and the initial boundary of EX and SH, as illustrated in the upper sketch of Fig. 11. For small depths of indentation, element $A$ is mainly subject to compression in the radial direction since the local stress field is dominated by the vertical compression from the indenter. As a result of Poisson ratio effects, the material in element $A$ expands in the circumferential direction. For small wedge angles, the contact length increases faster with load leading to a more rapid change in the position of the sector boundaries. In addition, a longer contact length leads to more constraint along the contact interface which restricts expansion closer to the indenter, i.e., with the increasing load, there is more constraint applied to element $A$ in the circumferential direction. The components of stress in the two directions achieve a balance state when the load is moderate, that is, the deformation of element $A$ is paused in this stage. After a short duration of equilibrium, the compressive stress in the circumferential direction becomes larger than that in the radial direction due to the indentation of the wedge and the corresponding rotation of material near contacting edge. Thus, the element $A, $ which is initially located in the EX sector, is transformed into the SH sector in this last stage. This seems to provide a reasonable hypothetical mechanism for the change in the size of the EX sector.

It should be noted that Refs.^[6]-[8] obtained asymptotic solutions for the strain fields associated with contact of a rigid wedge with a rubber notch assuming two-dimensional plane strain deformation. Therefore, these solutions are not directly applicable to the case examined here, where the rubber block is a plain rectangle initially, and there is considerable out-of-plane deformation which invalidates the plane strain assumption.

5 Conclusions

Stereoscopic DIC is used to measure the displacements resulting from indentation of a soft material block by a series of wedges with a round tip and various angles of slope. Experiments are performed with wedge angles that vary from sharp to blunt (73.5$^{\circ}$, 45$^{\circ}$, 15$^{\circ}$, and 5$^{\circ}$). The limits of validity for soft materials of the theory due to Ciavarella et al.^[10] are investigated by comparing measured and predicted in-plane displacement fields using image decomposition and found to be restricted to small indentation depths and small wedge angles. The classical blunt punch theory is also used for the smallest wedge angle, and the results are compared with both the predictions using the theory due to Ciavarella et al.^[10] and the results from the experiments. It is found that both the theories have a critical depth of indentation beyond which they are not valid, probably as a result of nonlinear behavior in the experiment, and this value can be evaluated quantitatively using the experimental analysis described above. The sectorial approach to modelling indented soft materials proposed by Refs.^[6]-[8] is verified by experiment and, the evolution of sectors during the indentation process is revealed for the first time.

Acknowledgements Xiaohua TAN was a recipient of China Scholarship Council Award\linebreak (No.2011625016) and E. A. PATTERSON was a recipient of Royal Society Wolfson Research Merit Award. The support of the University of Liverpool in providing the laboratory facilities used in this investigation is gratefully acknowledged.