Analytical model for elasto-plastic indentation of a hemispherical surface inclusion

An analytical model is developed to deduce the elastic and plastic properties of a hemispherical inclusion embedded in the surface of a semi-infinite solid from its indentation response. The model differs from the approaches presented in the literature by starting from the analytical expressions for the elastic and elasto-plastic indentation responses of homogeneous solids, and adapting them by replacing the modulus of the homogeneous solid by an effective modulus for the embedded inclusion. The accuracy of the indentation model is established by comparing the analytical results with detailed finite element simulations for various bi-material configurations of inclusion and substrate. The elastic indentation response is substantially influenced by the elastic modulus of the substrate, whereas the plastic response is dictated by the yield strength of the inclusion. The practical applicability of the indentation model is demonstrated by making use of the measured indentation response of an embedded paint sample, as reported in the literature, to deduce the elastic modulus of the paint.


Introduction
The indentation test is a widely used experimental technique for measuring material properties, such as elastic modulus [1,2], yield strength and hardness [3][4][5][6] and fracture toughness [7], over a wide range of length scales.The test entails the measurement of the force to press a shaped indenter tip into the surface of a sample; in principle, it is a simple test to perform but is notoriously difficult to interpret [8,9].The application of this technique at the micro and nano scales can make it possible to measure the mechanical properties (elastic modulus, strength and toughness) of small-scale material systems, such as thin coatings on a substrate [10,11], individual crystalline grains of a metallic microstructure [12][13][14], and small particles embedded in a supporting matrix [15,16], and can allow for the measurement of potential size effects [17][18][19][20][21]. Due to the local character of an indentation measurement, indentation testing is suitable for determining the properties of a material for which a limited number of (small) samples are available, for example paints that are used in valuable and historical paintings [22][23][24][25][26][27].
Paint samples subjected to indentation are typically embedded in a relatively stiff, supporting resin.Several methods have been proposed for obtaining the elastic modulus of embedded samples from experimental indentation data.One method aims at finding the maximum indentation depth such that the measured unloading modulus can be accurately computed without due influence by the embedding material [12,[28][29][30].Nevertheless, it is not always possible to limit the indentation depth to the small value required by this method, and consequently it is necessary to take into account the contribution to macroscopic compliance from the embedding material [15,24,[31][32][33].
Accurate analytical and numerical solutions have been obtained for the elastic indentation of diverse heterogeneous material systems, such as coatings supported by a semi-infinite substrate [34][35][36] and bi-materials that contain a vertical material interface [37].In [38] the elastic indentation of a hemispherical particle embedded at the free surface of a half space was studied, adopting the assumption that the indenter contact area remains relatively small with respect to the particle size.A first-order asymptotic solution of the Boussinesq-type problem was established, and the effect of the mismatch in elastic parameters on the indentation response was shown.However, much less is known about the relationship between Young's modulus and the macroscopic compliance of an embedded surface inclusion under relatively large indentations, as exhibited by indentation tests on embedded paint samples.
In the present study, an analytical model is derived to relate the elastic and plastic properties of an embedded hemispherical inclusion to its indentation response.The model differs from the approaches outlined above by starting from the analytical expressions for the elastic https://doi.org/10.1016/j.ijmecsci.2022.107267Received 31 January 2022; Received in revised form 30 March 2022; Accepted 31 March 2022 and elasto-plastic indentation responses of homogeneous materials, and adapting them by replacing the modulus of the homogeneous solid by an effective modulus for the embedded sample.The effective modulus is derived from the analytical solution for the elastic response of a spherical particle that contains a spherical cavity and is embedded in an infinite medium of different elastic properties.The cavity is subjected to a uniform, internal pressure.The accuracy of the analytical model is established by comparing the analytical indentation results for various bi-material configurations with those from detailed finite element simulations.The practical applicability of the indentation model is demonstrated by determining the elastic modulus of a paint from indentation measurements on an embedded paint sample as reported in the literature [24].
This paper is organized as follows.In Section 2 a review is provided of analytical elastic and elasto-plastic indentation models for homogeneous materials, and the accuracy of the analytical elastoplastic indentation model is assessed by comparing its response to that obtained from finite element simulations.In Section 3 an analytical expression is derived for the effective modulus of a bi-material.In Section 4 this analytical expression is combined with the analytical indentation models for homogeneous materials reviewed in Section 2 to simulate the indentation response of elastic and elastic, ideally plastic bi-materials.The accuracy of the analytical models is assessed by comparison with detailed finite element simulations.In Section 5 the practical applicability of the analytical indentation model is demonstrated by determining the elastic modulus of an embedded paint sample from experimental indentation data.Finally, the main conclusions of the study are presented in Section 6.

Review of indentation models for homogeneous materials
Similarity solutions have been developed for the indentation of a half space made from a power-law solid by a headshape of power-law form (such as a paraboloid or cone) [39,40].The force  applied to the indenter is related to the indentation depth ℎ and alternatively to the indentation contact radius  by [2,[41][42][43][44]  =  ℎ ℎ   and  =     , ( where , ,  ℎ and   depend upon the properties of the non-linear solid and upon the geometry of the headshape.
In general, the values of , ,  ℎ and   differ for loading and unloading, as unloading is usually associated with an elastic material response [45] 1 .The specific case of indentation of a linear elastic solid is treated in Section 2.1; elastic unloading of an indented elasto-plastic solid is discussed in Section 2.2, and indentation of an elastic, ideally plastic solid is considered in Section 2.3.

Indentation of an elastic solid
Consider a homogeneous, elastic half space indented by a frictionless, rigid conical punch.Harding [46] and Sneddon [1,2] showed that the force-displacement relations given by Eq. ( 1) are of the form where  is the semi-apex angle of the indenter and  and  are the Young's modulus and Poisson's ratio, respectively.The elastic compliance of the indenter can be accounted for by replacing the plane-strain elastic modulus ∕(1 − 2 ) in the above two expressions by a reduced modulus   defined as [3,41] 1 Fig. 1.Indentation of a homogeneous, elastic material over a depth ℎ, using an indenter with semi-apex angle .The contact radius, contact depth and sink-in displacement are given by , ℎ  , and ℎ  , respectively.
where   and   are the Young's modulus and Poisson's ratio of the indenter, respectively.Note from Eq. ( 2) that the exponents  and  in Eq. ( 1) have the value  =  = 2, and  ℎ and   are given by The geometrical parameters ℎ and  are related by As sketched in Fig. 1, sink-in occurs at the edge of the indenter to a depth ℎ  , as a result of which the contact depth associated with no sinkin (or pile-up), ℎ  = ∕ tan , is less than the true indentation depth ℎ, such that via Eq.( 5).The sink-in displacement ℎ  follows immediately from the identity as The initial value of the unloading contact stiffness (ℎ) is the derivative of Eq. (2) 1 with respect to the indentation depth ℎ: Note that, for an elastic indentation, the  (ℎ) curve is reversible and the loading slope equals the unloading slope .In contrast, for an elasto-plastic indentation the elastic unloading slope exceeds that of the elasto-plastic loading slope.Upon inserting Eq. ( 5) into Eq.( 9) and defining the projected contact area as   =  2 , the Young's modulus  is related to  via [41,43,47] The right-hand side of the above expression needs to be multiplied by a correction factor 1∕ when the projected contact area   is noncircular, with  equal to 1.012 and 1.034 for, respectively, square and triangular shape indents [48].

Elastic unloading of an elasto-plastic solid
Eq. ( 10) can be used to determine the Young's modulus from indentation tests on an elasto-plastic solid by following the procedure of Oliver and Pharr [41,43], summarized as follows.Assume that the indent is elasto-plastic in nature but unloading is elastic from an indentation depth ℎ = ℎ  to a residual depth ℎ  at zero load, see Fig. 2. Eq. (1) 1 is used to fit the unloading curve over the recoverable, elastic indentation interval (ℎ  − ℎ  ) as [41,43] As a result of the residual plastic strain distribution generated during loading, the value of the exponent  can be different from 2 (for conical indentation).The effect depends on the ratio between the Young's modulus  and yield strength   of the material, and thereby depends upon material class, see Table 1 in [45] for specific values measured from indentation tests on different materials.From the calibrated curve Eq. ( 11), the unloading contact stiffness  at the indentation depth ℎ = ℎ  is computed as Oliver and Pharr [43,45] further assumed that the sink-in displacement ℎ  is purely elastic; hence, for a conical indenter  equals 2, and ℎ  is related to the recoverable indentation interval (ℎ  − ℎ  ) in a manner analogous to Eq. ( 8), such that Additionally, by inserting the exponent  = 2 for elastic indentation in Eqs. ( 11) and ( 12), it follows that the initial elastic unloading stiffness where   is the indentation force at the indentation depth ℎ  , see Fig. 2. Now combine Eqs. ( 7), ( 13) and ( 14) to obtain with the unloading contact stiffness  of the elasto-plastic solid given by the general expression, Eq. ( 12), and  = 2( − 2)∕ = 0.73 for a conical indenter.A slightly larger value of  = 0.75 needs to be selected when the effective indenter shape during unloading is (accurately) approximated by a paraboloid of revolution [45].The projected contact area   is determined from the value of ℎ  as deduced from Eq. ( 15), using the fitting procedure described in [41].Alternatively, the contact area can be measured directly by imaging of the residual indent [41,43].Finally, the values of   and  are inserted into Eq.( 10) to obtain the Young's modulus  of the indented solid, while the hardness follows from  =   ∕  .

Indentation of an elastic, ideally plastic solid
Marsh [49] and Johnson [3,50] assumed that conical indentation of an elastic, ideally plastic solid by a rigid, conical indenter can be idealized by the expansion of an internally pressurized, hemispherical cavity from a vanishing initial radius to a current radius equal to the contact radius , see Fig. 3.The stress state  inside the void (or ''core'') is a uniform pressure,  = −, where  is the usual second-order identity tensor.The deformation state external to the hemispherical core is taken to be the same as that for an elastic, ideally plastic full space containing a spherical cavity of radius  and subjected to an internal pressure  [51].The plastic zone extends from the core to an outer elastic-plastic boundary at a radius  = , with  > .At the interface  =  between core and plastic zone, the radial stress in the plastic zone equals the hydrostatic stress  in the core.In addition, the radial displacement at the interface is compatible with the volume displaced by the conical indenter, assuming that the material within the core is incompressible.It is also assumed that no pile-up or sink-in occurs during indentation, such that the indentation depth is ℎ = ℎ  = ∕ tan , see Fig. 3.During loading, the load  is increased from zero to a value   , whereby the indentation ℎ obtains a value ℎ  .Under subsequent unloading, the indentation depth decreases towards a residual, plastic indent ℎ  at zero load.The reversible, elastic indentation experienced during unloading is (ℎ  − ℎ  ).The elastic contact stiffness at the onset of unloading is .This figure is based upon a representation presented in [41].Fig. 3. Cavity expansion indentation of a homogeneous, elastic, ideally plastic material to a depth ℎ, using an indenter with semi-apex angle .Indentation produces a hemispherical core of a radius equal to the contact radius .The stress generated inside the core induces a hemispherical plastic zone of radius .
Write   as the yield strength of the indented material and  = (90 • − ) as the inclination of the conical indenter.Then, the mean indentation pressure   under the indenter is given by [3,50]: where )) , Eqs. (16) 1 and (16) 2 define the elasto-plastic and fully plastic indentation responses, respectively.Note from Eq. ( 16) that the mean indentation pressure   is fully determined by the dimensionless parameter ( tan )∕  , which can be interpreted as the ratio of the representative strain imposed by the indenter, tan , and the yield strain,   ∕ [3].The use of alternative headshapes of indenters (spherical, Vickers, Berkovich) somewhat changes the point of first yield and full yielding in Eq. ( 16), but preserves the overall form of Eq. ( 16) [3].Although the solution given by Eq. ( 16) is based upon the assumption that the indented solid is incompressible, the effect of the value of Poisson's ratio upon the elastoplastic response is minor [3,50].The compliance of the indenter is taken into account by replacing the Young's modulus  in Eq. ( 16) by the reduced modulus   as given by Eq. ( 3) [3,50].The force  on the conical indenter follows directly from the mean indentation pressure   and the projected contact area   = 2 , such that  =     , (17) and consequently the constant   in the general expression, Eq. ( 1) 2 , specializes to   =   , with   given by Eq. ( 16), and the exponent  = 2. Further, from the relation ℎ = ∕ tan , the parameters  ℎ and  in Eq. ( 1) 1 follow as  ℎ =   (tan ) 2 and  = 2. Three-dimensional indentation simulations reveal that Johnson's cavity expansion model gives a rather good representation of the nominal hardness of polymers (e.g., paints); an approximately spherical plastic zone develops and no material pile-up occurs next to the indenter [52,53].In order to verify the accuracy of Eq. ( 16), the results of this model are compared to large deformation Finite Element (FE) simulations of the indentation of an elastic, ideally plastic half space, using the commercial FE package ABAQUS Standard 2 .The indentation problem is modelled as axisymmetric, with the vertical line of symmetry passing through the centre of the indenter.
The indented solid is discretized using axisymmetric, 4-node, isoparametric elements with 2 × 2 Gaussian quadrature.The conical indenter is modelled as rigid and frictionless, and is characterized by a semi-apex angle of  = 70.3• , corresponding to an inclination  = 19.7 • of the indenter.The numerical stability of the solution is enhanced by slightly rounding off of the indenter tip with a small, finite tip radius.The inclusion of a tip radius also makes the indenter more representative of a practical conical indenter.The indenter is displaced vertically into the solid using an incremental time-marching scheme with an automatic time-step adaptation.The maximum indentation depth is set to 8 μm, which is a factor of 6.25 smaller than the radial and vertical dimensions of 50 μm defining the axisymmetric finite element configuration.A so-called node-to-surface contact criterion is adopted in order to rigorously check for new contacts between the indenter and the elements that define the top surface of the solid.
The FE mesh is refined near the indenter tip, and the semi-infinite character of the solid is simulated by placing 4-node infinite elements with an elastic material behaviour along the lateral and lower boundaries of the finite element geometry [54].The converged FE discretization deduced from the above mesh refinement study corresponds to a spatial discretization of 4499 finite elements and 100 infinite elements.The choice of element discretization is determined from a preliminary mesh refinement study on the initial, elastic indentation response.In the mesh refinement study the elastic response was simulated by selecting an almost incompressible solid with  = 0.499; the converged numerical results agree with the analytical solution given by Eq. ( 2) to within an acceptable inaccuracy of 1%.
In the FE simulations of the indentation of the elastic, ideally plastic solid the mesh density of the converged elastic solution is preserved and the size of the mesh is increased by a factor of 10 in the radial and vertical directions to ensure that the plastic zone generated by the conical indenter does not reach the elastic infinite elements located at the perimeter of the FE model.Consequently, the number of finite elements in this FE model equals 15 557.The number of infinite elements is kept the same as in the mesh convergence study, that is 100.The Poisson's ratio of the solid  = 0.35 is representative of various solids, including historical paints [55]. 2 -flow theory is used for the plastic response, and the Young's modulus  and yield strength   are selected such that the dimensionless parameter ( tan )∕  appearing in Eq. ( 16) is varied stepwisely from 1 to 100, corresponding to 100 separate FE simulations.Fig. 4 shows a comparison between the results obtained from the FE simulations (grey circles and black solid line) and Johnson's cavity expansion model, Eq. (16) (dashed line).Each vertical column of data points, designated by the grey circles, corresponds to an increase in indentation depth ℎ for an integer number of contact nodes.The bottom point of each column of data points gives the instant at which a new node comes into contact with the indenter.Thus, the contact radius  remains constant and the mean indentation pressure   (plotted along the vertical axis) grows until the next node makes contact.This feature of mesh discretization repeats itself and a new column of vertical data points is generated, see also [56][57][58].The black solid line depicted in Fig. 4 captures the mean values of the vertical columns of data points.The cavity expansion model, Eq. ( 16), is in reasonable agreement with the response of the FE simulations, with an underprediction in the intermediate elasto-plastic regime and an overprediction in the final, plastic regime.Although not illustrated here, the FE results also show that the ratio of contact depth ℎ  to indentation depth ℎ in the elastoplastic regime monotonically increases from ℎ  ∕ℎ = 2∕ to ℎ  ∕ℎ = 1: the initial, elastic value is in agreement with the expression given by Eq. (6).When ( tan )∕  exceeds 27, the plastic deformation regime is reached, whereby the mean indentation pressure   becomes constant, and the contact depth develops with ℎ  ∕ℎ > 1, indicating that material piles up next to the indenter, as previously discussed [3,50].
Additional comparisons between the results of the cavity expansion model and those of FE simulations can be found in [50], including a comparison of the subsurface and surface stress fields in Figures 6.15 and 6.16, which shows good agreement.

Closed-form expression for the effective elastic modulus of a bi-material
The indentation models for a homogeneous solid reviewed in Section 2 will be adapted to the case of a bi-material composed of a hemispherical inclusion at the surface of a dissimilar half space, with the indent placed at the centre of the inclusion.As will be demonstrated in the current section and Section 4, this adaptation will be achieved via the derivation of an effective elastic modulus of the bi-material, followed by taking this effective modulus as the value of Young's modulus in the indentation models for a homogeneous solid.In the present section an analytical expression for the effective elastic modulus is derived from the response of a hollow, spherical particle embedded in an infinite medium of dissimilar elastic properties, and subjected to a uniform, internal pressure, see Fig. 5.The elastic displacement field in the bi-material is derived in Section 3.1, and an analytical expression for the effective elastic modulus is obtained in Section 3.2.Fig. 5.A spherical shell of outer radius  (material 1) embedded in an infinite medium (material 2).The spherical cavity of radius  is pressurized by a uniform pressure .The inset shows the spherical coordinate system (, , ), with  ≥  the radial coordinate of a material point,  the polar angle coordinate, and  the azimuth angle coordinate.

Elastic displacement field
The geometry depicted in Fig. 5 is characterized by an elastic spherical shell of outer radius  (material 1) that is embedded in an infinite solid of dissimilar elastic properties (material 2).The inner, spherical cavity of radius  is subjected to an internal uniform pressure .A spherical coordinate system (, , ) is adopted, where  ≥  is the radial coordinate,  the polar angle coordinate, and  is the azimuth angle coordinate.Upon exploiting the spherical symmetry of the problem, the following equilibrium and kinematic conditions hold: Here,   ,   and   are the stress, strain and displacement components in the spherical coordinate system.The constitutive relations and the remaining equilibrium and kinematic relations of the solid are where   and   are the Young's modulus and Poisson's ratio of material  ∈ {1, 2}.Additionally, the boundary conditions (at  =  and  → ∞) and interfacial conditions (at  = ) are given by: in which the comma and subindices 1 and 2 used in the definition of the interfacial conditions Eq. (20) 3,4 refer to materials 1 and 2, respectively.Combining Eqs. ( 19) and ( 20) leads to the following expression for the radial displacement: for  ≤  ≤ , For the specific case of a vanishing material 2 (i.e.,  2 = 0,  2 = 0), Eq. ( 21) reduces to: in which the subscripts of the elastic parameters  1 and  1 have been dropped for the sake of clarity.Eq. ( 22) gives the elastic displacement field for a homogeneous spherical shell of finite thickness (−), with a free outer boundary at  = , and loaded by a uniform pressure  at its inner boundary  = ; this expression is in agreement with that given in [51].For the special case of an internally pressurized spherical cavity in an infinite, homogeneous medium, i.e.,  1 =  2 =  and  1 =  2 = , Eq. ( 21) reduces to the classical solution [59]:

Effective elastic modulus of an embedded spherical shell
It follows from Eqs. (19) 4 and ( 23) that, for the case of a homogeneous infinite medium, the radial strain at the boundary  =  of the cavity can be written as which, upon rearrangement, expresses the Young's modulus as a function of the radial strain at  = : The assumption is made that the embedded spherical shell can be idealized by an equivalent cavity of radius  in a homogeneous full space by introducing an effective modulus Ē for the cavity in a homogeneous full space.The calibration is based on the assumption that the radial strain at the boundary  =  of the cavity is the same for the two cases.Thus, the effective modulus Ē for the cavity in a homogeneous full space is defined by rearrangement of the above equation to read: Here, the value of the radial strain at the boundary  =  is for the embedded spherical shell, and is determined from Eqs. (19) 4 and (21) 1 as Now insert Eq. ( 27) into Eq.( 26) to obtain the effective elastic modulus: This expression reduces to the Young's modulus of a homogeneous material, Ē = , when  1 =  2 =  and  1 =  2 = , as required.In Section 4 the above expression for the effective elastic modulus of the bi-material is combined with the indentation models for homogeneous materials reviewed in Section 2 to obtain an analytical expression for the indentation response of elastic and elasto-plastic bi-materials.
Note that an alternative expression for the effective modulus of the bi-material can be obtained by suitable matching of the hoop strain of the bi-material problem and the cavity in an effective, homogeneous full space at  = .Combine Eqs.(19) 5 and ( 23) for the cavity in a homogeneous full space, and substitute  by Ē and  by  1 : Fig. 6.Indentation of an elastic hemispherical particle of radius  (material 1) embedded in an elastic half space (material 2), using an indenter of semi-apex angle .The contact radius , contact depth ℎ  and sink-in depth ℎ  each depend upon the indentation depth ℎ.
Then, assume that the value of   ( = ) for the embedded spherical shell equals that for the equivalent effective full space.A comparison study has been performed between the use of hoop and radial strain components to obtain Ē, but is not detailed here.It is found that the effective modulus Ē based on Eq. ( 26) provides a more accurate estimate of the elastic indentation response of an embedded hemispherical inclusion than that based on Eq. ( 29).

Indentation of embedded hemispherical particles
The closed-form expression for the effective elastic modulus of the bi-material, Eq. ( 28), is now combined with the analytical indentation models for homogeneous materials, as reviewed in Section 2. First, the indentation of an elastic hemispherical particle embedded in an elastic half space is reported in Section 4.1.Second, the indentation of an elastic, ideally plastic hemispherical particle embedded in an elastic half space is given in Section 4.2.In each section, the accuracy of the analytical models is established by comparing analytical predictions with detailed finite element simulations.

Indentation of an elastic hemispherical particle embedded in an elastic half space
The elastic indentation of a hemispherical particle embedded in a half space of differing elastic properties is sketched in Fig. 6.The analytical indentation model makes use of Eq. ( 28) in the  − relation, Eq. (2) 2 .Accordingly, the indentation force  on the rigid conical indenter is expressed in terms of the contact radius  of the bi-material as in which the Poisson's ratio corresponds to that of the indented material 1. Eq. ( 30) has been extended by a factor , which corrects for a small, artificial overlap between the geometry of the rigid indenter and the deformed material surface, as characteristic of Sneddon's solution, see also [60].An analytical expression for the factor  is obtained by calibrating Eq. ( 30) on the FE indentation response of a homogeneous material -here designated as ''material 1'' -with the Poisson's ratio taking values in the range of 0 to 0.5.The FE mesh used for the axisymmetric indentation models corresponds to that from the mesh convergence study described in Section 2.3.The calibration of the factor  is performed when the indentation response has converged towards a steady state, as characterized by an almost constant value of the dimensionless indentation force  ∕(   1 ), with   =  2 the projected contact area.The value of  is taken as the average of the Fig. 7. Analytical expression for the factor  (black line) as given by Eq. ( 31), which is obtained from calibrating Eq. ( 30) on the FE indentation response of a homogeneous, elastic half space (red circles) for selected values of Poisson's ratio  1 .
values calibrated for a range of contact radii .Fig. 7 illustrates that the numerical values for  are accurately captured by the linear relation Note that  = 1 in the limit of an incompressible material,  1 = 0.5, which is the case that has been selected for the mesh refinement study discussed Section 2.3.

Elastic indentation response
The elastic indentation response of the bi-material configuration sketched in Fig. 6 is illustrated in Fig. 8. Specifically, the dimensionless indentation force  ∕(   1 ) is plotted as a function of the dimensionless indentation radius ∕ for three values of modulus mismatch,  2 ∕ 1 = [0.2,1.0, 5.0], and 6 choices of Poisson's ratios,  1 =  2 in the range of 0 to 0.5.The black solid line represents the FE response, which develops in a sawtooth fashion due to the fact that the contact condition of the indenter is prescribed in a spatially discrete manner via the individual finite element nodes, see also Fig. 4 and the explanation provided in Section 2.3.The mean of the FE sawtooth curve is given by the red dotted line.The analytical model, Eq. ( 30), with Ē and  respectively given by Eqs. ( 28) and (31), is represented by the black dashed line.Additionally, for the specific case of  2 ∕ 1 = 0.2, the analytical model with an effective modulus for an incompressible bi-material is depicted in Figs.8(a)-(f) by the black dotted line; the corresponding expression for the effective modulus follows from inserting  1 =  2 = 0.5 into Eq.( 28): with the subscript ''" referring to ''incompressible".This expression correctly reduces to Ē =  1 for a homogeneous material,  2 ∕ 1 = 1.0.The analytical indentation model, Eq. ( 30), that uses the effective modulus of the incompressible material, Eq. ( 32), henceforth will be denoted ''analytical model for the incompressible bi-material''; the indentation model also describes indented materials with a Poisson's ratio  1 different from 0.5, via the terms  = 1.2 − 0.4 1 and (1 −  2 1 ) in Eq. (30).
For the case  2 ∕ 1 = 0.2 of a stiff particle embedded in a soft matrix, the mean FE results plotted in Fig. 8 show that the dimensionless indentation force  ∕(   1 ) monotonically decreases with increasing indentation radius ∕ after a relatively short initiation phase 3 .In contrast, for a homogeneous material,  2 ∕ 1 = 1.0,  ∕(   1 ) is G.J.A.M. Eumelen et al.Fig. 8. Indentation response of an elastic hemispherical particle embedded in an elastic half space.Dimensionless indentation force  ∕(   1 ) versus dimensionless indentation radius ∕, as calculated by FE simulations (black solid line), the analytical model Eq. ( 30) with the general stiffness expression Eq. ( 28) (black dashed line), and the analytical model Eq. ( 30) with the stiffness expression for an incompressible material Eq. ( 32) (black dotted line).The mean FE response is represented by the red dotted line.The results are shown for three values of modulus mismatch,  2 ∕ 1 = 0.2, 1.0, and 5.0, and for (a) independent of ∕, and for the case  2 ∕ 1 = 5.0 of a compliant particle embedded in a stiff matrix  ∕(   1 ) increases monotonically with ∕.Remarkably, the indentation load for the stiff particle configuration,  2 ∕ 1 = 0.2, is almost insensitive to the choice of Poisson's ratios  1 and  2 , and is adequately captured by the analytical model for an incompressible bi-material, Eq. ( 32).The analytical model with the general stiffness expression, Eq. ( 28), overpredicts the FE results for this case, although the discrepancy decreases for a larger value of Poisson's ratios, and eventually vanishes in the limit of an incompressible bimaterial,  1 =  2 = 0.5.For a homogeneous material,  2 ∕ 1 = 1.0, the analytical models with the stiffness expressions given by Eqs. ( 28) and (32) lead to an identical result, and accurately describe the mean FE response.For the compliant particle configuration,  2 ∕ 1 = 5.0, the analytical model with the general stiffness expression Eq. ( 28) gives close agreement with the mean FE results over the full range of dimensionless indentation radii ∕ in case of moderate values of the Poisson's ratios, 0.2 ≤   ≤ 0.3 and  ∈ {1, 2}.Since this range includes the Poisson's ratios of many engineering materials, it is concluded that this analytical indentation model is of practical value.For values of Poisson's ratios falling outside this range, Fig. 8 shows that the analytical model only provides accurate results up to a normalized indentation radius of ∕ ≈ 0.3 to 0.4, and for larger indentation values may significantly underpredict (for  1 ,  2 < 0.2) or overpredict (for  1 ,  2 > 0.3) the mean FE response of the compliant particle configuration.Finally, note from Fig. 8(f) that, for  2 ∕ 1 = 5.0, the analytical model for the incompressible bi-material deviates from the FE results obtained for a Poisson's ratio of 0.5, which is the reason that analytical predictions for an incompressible bi-material with  2 ∕ 1 = 5.0 have been omitted from Figs. 8(a)-(e).

Effective elastic modulus
The mean FE results indicated by the red dotted line in Fig. 8 can be used to compute the normalized effective elastic modulus Ē∕ 1 of the bi-material from the inverse relation of Eq. ( 30), i.e., Accordingly, in Figs.9(a)-(f) the effective modulus following from the FE simulations is compared to the effective modulus, Eq. ( 28), and the effective modulus for the incompressible bi-material, Eq. ( 32), by plotting the dimensionless value Ē∕ 1 against the dimensionless indentation radius ∕ for three values of stiffness mismatches  2 ∕ 1 = [0.2,1.0, 5.0] and six values of Poisson's ratio over the range 0 to 0.5.
After a minor initiation phase, all curves attain an effective modulus of Ē =  1 , which confirms that at small indentation the effective modulus equals the Young's modulus of the hemispherical particle.Under continued indentation, the effective modulus of the stiff particle configuration,  2 ∕ 1 = 0.2, monotonically decreases, for the homogeneous material,  2 ∕ 1 = 1.0, it remains constant and equals Ē =  1 , and for the compliant particle configuration,  2 ∕ 1 = 5.0, it monotonically increases.In agreement with the observation made from Fig. 8, for the stiff particle configuration,  2 ∕ 1 = 0.2, the analytical model for the incompressible bi-material closely matches the effective modulus determined from the FE simulations.For the homogeneous material the general analytical expression, Eq. ( 28), and the expression for the incompressible bi-material, Eq. ( 32), lead to the same result Ē =  1 , which match the FE results.For the compliant particle configuration,  2 ∕ 1 = 5.0, the agreement between the effective modulus expression, Eq. ( 28), and the FE results is adequate for Poisson's ratios in the range 0.2 ≤   ≤ 0.3 where  ∈ {1, 2}, and is adequate up to an indentation of ∕ = 0.3 to 0.4 for Poisson's ratios falling outside this range.
From the above comparison with the FE results (and from similar comparisons with FE results for alternative stiffness mismatches  2 ∕ 1 = 0.5 and 2.0, but omitted here for the sake of brevity), it is concluded that the analytical expressions for the effective elastic modulus Ē of the indented bi-material given by Eqs. ( 28) and ( 32) are adequate for a broad range of stiffness mismatches, 0.2 ≤  2 ∕ 1 ≤ 5.0, and Poisson's ratios, 0 ≤   ≤ 0.5 with  ∈ {1, 2}, within the following regimes of the normalized indentation radius ∕: , with   ∈ { 1 ,  2 }.In the above expressions the various regimes follow each other through the specific ranges selected for the elastic parameters.The homogeneous limit  2 ∕ 1 = 1.0 is omitted from Eq. ( 34), but, as has already been mentioned, the correct limit is obtained from this equation as Ē =  1 for the complete range of Poisson's ratios 0 ≤   ≤ 0.5.Further, the maximum indentation radius ∕ = 1 for simplicity has been included in the applicability ranges of Eq. ( 34), although for certain combinations of elastic parameters the maximum value of ∕ may be somewhat smaller than unity, due to the fact that the indenter prematurely touches the supporting material 2 under a relatively large vertical deformation of material 1, see also Figs. 8 and  9.

Indentation of an elastic, ideally plastic hemispherical particle embedded in an elastic half space
The elastic configuration studied in Section 4.1 is now extended to the case of an elastic, ideally plastic hemispherical particle (material 1) embedded in an elastic half space (material 2), see Fig. 10.The size of the plastic zone generated within the hemispherical particle is denoted by the radius .The plastic zone can expand until it reaches the material interface between the hemispherical particle and the supporting elastic half space,  = , or until the indenter reaches the material interface,  = .The indentation response of the bi-material configuration is analysed analytically by an adapted version of Johnson's cavity expansion model for a homogeneous material Eq. ( 16), and numerically by means of finite element analyses.The adaptation of Johnson's model involves the replacement of the Young's modulus  in Eq. ( 16) by the analytical expression for the effective elastic modulus Ē of the bi-material, leading to where with Ē given by Eq. (28).Instead of Eq. ( 28), the refined, more complicated stiffness expression, Eq. ( 34), could have been used, but this leads to only minor differences in the elasto-plastic indentation response.Fig. 11 illustrates the indentation response from both the FE simulations and the analytical cavity expansion model of the bi-material, Eq. (35).The Poisson's ratios of the materials 1 and 2 are taken to be  1 = 2 =0.35.Note that the incorporation of the effective elastic modulus Ē of the bi-material in the dimensionless parameter ( Ē tan )∕  plotted along the horizontal axis is consistent with the analytical expression in Eq. (35).Consequently, the response plotted for the cavity expansion model includes the role of stiffness mismatch  2 ∕ 1 .As already discussed in Section 2.3, the mean FE response for a homogeneous material, Ē =  1 =  2 , is adequately approximated by the cavity expansion model.The mean FE responses for the bi-material refer to a high stiffness ratio  2 ∕ 1 = 10.0 (red solid line) and to a low stiffness ratio  2 ∕ 1 = 0.1 (blue solid line).These FE responses are calculated by performing separate analyses for 9 selected values of yield strength   , which start along the horizontal axis at 9 selected values of ( Ē tan )∕  .In correspondence with the range of validity of the cavity expansion model, the FE analyses are continued until the plastic zone reaches the material interface,  = , or the indenter reaches the material interface,  ≈ .
For small indentations the contribution of the supporting material 2 to the overall response is negligible, as a result of which the effective modulus equals that of the hemispherical particle, Ē =  1 .Under increasing indentation of the particle embedded in a relatively compliant Fig. 9. Indentation response of an elastic hemispherical particle embedded in an elastic half space.Dimensionless effective elastic modulus Ē∕ 1 versus dimensionless indentation radius ∕, as calculated by FE simulations and using Eq. ( 33) (black solid line), the analytical model Eq. ( 28) (black dashed line), and the analytical model for the incompressible material Eq. ( 32) (black dotted line).The results are shown for three values of modulus mismatch,  2 ∕ 1 = 0.2, 1.0, and 5.0, and for (a) half space,  2 ∕ 1 = 0.1, the values of ( Ē tan )∕  and   ∕  decrease in the initial elastic regime and also in the subsequent elasto-plastic regime.Specifically, the contribution of the compliant half space to the response grows with increasing indentation, so that the values of the mean hydrostatic stress   =  ∕  and the effective modulus Ē diminish, as already observed for the elastic responses shown in Figs. 8  and 9, respectively.For the same reason, the elastic and elasto-plastic indentation responses of the particle embedded in a relatively stiff half space,  2 ∕ 1 = 10.0, reveal an increase of ( Ē tan )∕  and   ∕  .Note further that, in the final, plastic regime, the mean indentation pressure   is almost insensitive to the elastic properties of the bi-material, and is essentially set by the yield strength   .
In summary, the elastic, ideally plastic indentation response of a bi-material can be quantified in terms of the effective elastic modulus, Eq. ( 28), in the representation in Fig. 11, with the FE results supporting the analytical cavity expansion model, Eq. (35).Thus, the cavity expansion model can be used as a practical, analytical tool to estimate the elasto-plastic material properties of embedded samples from indentation tests.

Case study: use of analytical model to interpret indentation test on embedded paint sample
The practical applicability of the analytical expression for the effective modulus, Eq. ( 28), is demonstrated through a case study on Fig. 10.Indentation of an elastic, ideally plastic hemispherical particle of radius  (material 1) embedded in an elastic half space (material 2), by an indenter of semiapex angle .The indentation produces a hemispherical core of radius equal to the contact radius , and a hemispherical plastic zone of radius .Fig. 11.Indentation of an elastic, ideally plastic hemispherical particle embedded in an elastic half space.Mean indentation pressure   normalized by the yield strength   as a function of the dimensionless parameter ( Ē tan )∕  , with Ē given by Eq. (28).The FE results are depicted for a homogeneous solid,  2 ∕ 1 = 1.0 (black solid line), and for bimaterials with modulus mismatches  2 ∕ 1 = 0.1 (blue solid line) and  2 ∕ 1 = 10.0 (red solid line), with 9 initial values of ( Ē tan )∕  .The Poisson's ratios are  1 =  2 = 0.35.The analytical result for the cavity expansion model of the bi-material Eq. ( 35) (black dashed line) holds for arbitrary value of the stiffness mismatch  2 ∕ 1 via the parameter Ē.
indentation tests taken from the literature [24].Indentation tests were performed on an acrylic, titanium white paint layer of Golden Artist Colors ® (material 1) that was fully embedded in a resin specified as a Technovit ® 2000 LC fixing paste (material 2).The width of the paint sample was 210 μm and the thickness in the depth direction was considerably larger, above 1 mm.The reduced modulus measured by indenting the resin with a diamond Berkovich indenter was   ≈ 15 GPa [24].Assuming a representative value of  2 = 0.35 for the Poisson's ratio of the resin, Eq. (3) implies that  2 ≈   (1 −  2  2 ) = 13.2GPa.The challenge is to deduce the Young's modulus  1 of the paint from the measured unloading modulus  of an indentation test on the paint layer.
A sketch of the geometry at the material interface and the characteristics of the indenter is presented in Fig. 12.The embedded paint sample was subjected to two indentations, performed at distances  = 12 μm and  = 72 μm from the vertical interface between the paint and the resin.The indentation tests were carried out using a continuous stiffness measurement (CSM) technique, entailing harmonic loading of amplitude  0 at a frequency of 20 Hz, superimposed on the quasi-static Fig. 12. Indentation of a semi-infinite elastic bi-material with a straight, vertical material interface.The indentation of material 1 (paint) is performed at a distance  from the interface with material 2 (resin).The indentation depth ℎ relates to a contact radius , a contact depth ℎ  , and a sink-in depth ℎ  .The configuration is representative of the embedded paint sample tested in [24].loading.This allows for the determination of the contact stiffness as a function of indent depth without interrupting the indentation [8,43].The amplitude ℎ 0 of the induced dynamic oscillation was on the order of nanometres, much less than that of the quasi-static signal.The unloading contact stiffness  equals ( 0 ∕ℎ 0 ) cos , where  is the phase shift between the harmonic load and displacement induced in the embedded viscoelastic paint sample [8,[61][62][63].
The contact stiffness  measured at a specific quasi-static loading step   and corresponding indentation depth ℎ  were determined from the raw experimental indentation data 4 , and provide the actual contact depth ℎ  through Eq. ( 15) (using  = 0.75), which, under the assumption of a perfect Berkovich indenter, subsequently renders the contact area as   =  2 = (ℎ  tan ) 2 [41], where  = 68.6 • for the Berkovich indenter employed in the experiment [24].From the value of   and the corresponding value of , the effective modulus Ē of the bi-material is calculated as Eq. ( 36) is an approximation based on Eq. ( 10) for a homogeneous solid, whereby the Young's modulus  is replaced by the effective modulus Ē.Further, the factor  given by Eq. ( 31) has been added to correct for the artificial overlap between the indenter geometry and the deformed material surface, which typifies Sneddon's solution, Eq. ( 10).The accuracy of Eq. ( 36) for determining the effective modulus of the experimental bi-material configuration sketched in Fig. 12 has been confirmed in the Appendix via a comparison with the indentation result obtained from a detailed, three-dimensional FE simulation.Since the Berkovich indenter used in the experiment generates a triangular shape indent, the effective modulus computed with Eq. ( 36) needs to be multiplied by a correction factor 1∕, with  = 1.034, recall the discussion below Eq. (10).Figs.13(a) and (b) show the normalized effective modulus, Ē∕ 1 , as a function of the normalized contact radius ∕ of the indenter, as measured at distances  = 12 μm and  = 72 μm from the material interface, respectively.The grey circles represent the experimental data in the format provided by the stiffness expression, Eq. ( 36), and the black solid line reflects the result from the analytical model, Eq. ( 28).In both figures the experimental data is accurately matched by the analytical for the helpful digressions on their indentation experiments of embedded paint samples and for providing the raw experimental data.A.S.J.S. gratefully acknowledges the support of the European Union's Horizon 2020 research and innovation programme CollectionCare, under grant agreement N. 814624.E.B. is grateful for the support by the Netherlands Organization for Scientific Research (NWO), Project 15873, ''Engineering goes Beauty -A computational multi-physics modelling approach towards the preservation of historical oil paintings'', within the funding scheme ''NWO Veni Award''.N.A.F. is grateful for funding from the European Research Council (MULTILAT, grant number 669764).

Appendix. Three-dimensional finite element model versus analytical model for a paint indentation problem
The bi-material configuration sketched in Fig. 12 represents a paint (material 1) embedded in a resin (material 2), which has been subjected to indentation testing as reported in [24], see also Section 5.The suitability of the analytical model, Eq. ( 28), for determining the effective stiffness Ē of this configuration is assessed by means of a comparison with the results obtained from a detailed 3D FE indentation model.The 3D FE simulation is carried out in a similar fashion as described in Section 2.3 for the axisymmetric indentation models.For the tests performed in [24] the sample depth and width are considerably larger than the indentation contact radius  and the distance  between the indenter and the material interface; consequently, the geometry of the embedded sample is treated as semi-infinite.The origin of the (, , ) coordinate system shown in Fig. 12 corresponds to the horizontal centre point of the half space configuration, and is located along the free upper boundary, at a distance  from the vertical interface between the paint (material 1) and the supporting resin (material 2).The dimensions of the FE geometry are 100 × 100 × 50 μm 3 .This tetragonal volume is discretized using 8-node iso-parametric brick elements with a 2 × 2 × 2 Gaussian quadrature.The model symmetry in the -direction is exploited by applying appropriate fixed and roller supports along the  −  plane that crosses the origin of the (, , ) coordinate system.The semi-infinite character of the half space is simulated by placing 8-node infinite elements along the lateral boundaries and the lower boundary of the FE model.The FE mesh is constructed by employing 606 786 finite elements and 22 594 infinite elements, whereby the mesh density is increased towards the indenter tip for obtaining highly accurate numerical results, as confirmed from a mesh refinement study.
Fig. 14 shows the effective modulus calculated with the FE model and the analytical model, by plotting the dimensionless modulus Ē∕ 1 versus the dimensionless indentation radius ∕ for a stiffness mismatch of  2 ∕ 1 = 5.0 and Poisson's ratios of  1 =  2 = 0.35.The value of the stiffness mismatch is considered as an initial estimate for the embedded paint sample tested in [24].The effective modulus is deduced from the FE results by applying the stiffness expression Eq. ( 36), whereas for the analytical result it directly follows from Eq. (28).The unloading contact stiffness  in Eq. ( 36) is determined from the FE results via the discrete approximation of its definition given in Eq. ( 12), i.e.,  ≈  ∕ℎ, with  and ℎ the numerical increments of the applied load and the indentation depth, respectively.The results of the FE simulation are plotted up to the stage at which the indenter reaches the supporting material 2; as indicated in Fig. 12, the contact radius  is measured in the direction opposite to the material interface, so that the indenter reaches the material interface at a value ∕ somewhat smaller than unity, i.e., ∕ = 0.76.It can be observed in Fig. 14 that the analytical effective modulus matches the numerical effective modulus closely over its full range of indentation radius ∕.Hence, the analytical model, Eq. ( 28), can be used for an accurate calibration of the experimental indentation response presented in [24] if the interpretation of the test data is done with the stiffness expression, Eq. (36).
Although the initially estimated stiffness mismatch of  2 ∕ 1 = 5.0 adopted for generating the results in Fig. 14 is somewhat lower than the stiffness mismatch of  2 ∕ 1 = 8.3 deduced in Section 5 from the experimental indentation data, it may be reasonably expected that this does not significantly affect the accuracy of the calibration procedure.

Fig. 2 .
Fig.2.Schematic representation of the loading and unloading stages in an indentation experiment.During loading, the load  is increased from zero to a value   , whereby the indentation ℎ obtains a value ℎ  .Under subsequent unloading, the indentation depth decreases towards a residual, plastic indent ℎ  at zero load.The reversible, elastic indentation experienced during unloading is (ℎ  − ℎ  ).The elastic contact stiffness at the onset of unloading is .This figure is based upon a representation presented in[41].

Fig. 4 .
Fig. 4. Indentation response of an elastic, ideally plastic solid.Mean indentation pressure   normalized by the yield strength   as a function of the dimensionless parameter ( tan )∕  .The figure shows the FE response (grey circles), the mean FE response (solid black line) and the prediction by the cavity expansion model, Eq. (16) (dashed line).

Fig. 14 .
Fig. 14.Indentation response of the semi-infinite elastic bi-material with a straight, vertical material interface as shown in Fig. 12 (which represents an embedded paint sample).The modulus mismatch equals  2 ∕ 1 = 5.0 and the Poisson's ratios are  1 =  2 = 0.35.Dimensionless effective elastic modulus Ē∕ 1 versus the dimensionless indentation radius ∕, as determined from applying the stiffness expression Eq. (36) (solid line) to the FE results, and from the analytical model Eq.(28) (dashed line).