An integral approach of indentation of Functionally Graded Materials

Journal Pre-proof An integral approach of indentation of Functionally Graded Materials

Thierry Coorevits, Alberto Mejias, Alex Montagne, Stephania Kossman, Alain Iost PII:

S0257-8972(19)31166-1

DOI:

https://doi.org/10.1016/j.surfcoat.2019.125176

Reference:

SCT 125176

To appear in:

Surface & Coatings Technology

Received date:

24 June 2019

Revised date:

23 October 2019

Accepted date:

17 November 2019

Please cite this article as: T. Coorevits, A. Mejias, A. Montagne, et al., An integral approach of indentation of Functionally Graded Materials, Surface & Coatings Technology (2019), https://doi.org/10.1016/j.surfcoat.2019.125176

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© 2019 Published by Elsevier.

Journal Pre-proof An integral approach of indentation of Functionally Graded Materials Thierry Coorevits1, Alberto Mejias1, Alex Montagne1, Stephania Kossman1, Alain Iost1 1

Arts et Métiers ParisTech, MSMP EA-7350, 59800 Lille, France

[email protected]; [email protected]; [email protected]; [email protected]; [email protected]

Abstract. Determination of hardness profiles of functionally graded materials could be currently

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performed by two principal methods. One method consists in the measurement of hardness on the

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cut cross-section of the sample, which is destructive, expensive and time consuming. Another

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method involves the measurement of hardness at the surface with increasing loads. This is an inverse problem that is usually solved in indentation by multilayer models. Our aim is to generalize the

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different multilayer models developed in recent years from geometrical considerations. This proposal

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is based on an integral approach of the centroid model proposed by Jönsson and Hogmark (JH) (1984). A brief calculation gives a definite integral relating the composite hardness obtained by

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normal indentation tests and the pressure exerted on a section of the indenter. The reasoning of JH

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model gives the cross-sectional hardness as a function of the distance from the top surface, i.e. the hardness profile, which is calculated thanks to the definition of the mechanical pressure exerted on a section of the indenter. Then, an inverse problem must be solved. Two gradient models are proposed and tested on data of previous works, the Gauss error function (erf function) that models diffusion phenomena, and a linear gradient model in a piecewise-defined function. In these models, the influences of the indentation size effect and of the residual stresses are not considered. An important application of this work is the prediction of the hardness profile of mechanical parts subjected to thermochemical treatments (carburizing, nitriding, carbonitriding, among others) or deposited coatings when indentation tests are carried out on the normal surface of the material. Hardness profile calculated from normal hardness measurements at the surface by using the proposed approach could provide an appropriated hardness description of the sample cross-section.

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Journal Pre-proof Keywords: Hardness modelling; hardness profile; coating; indentation; Gaussian statistics; Monte Carlo method. 1

Introduction

Mechanical properties of materials are obtained by different standardized methods [1, 2, 3, 4]. Hardness, Young’s modulus and fracture toughness are among the mechanical properties commonly evaluated by these methods for which a bulk material sample is used. However, when the samples are too small to evaluate the mechanical properties by conventional tests, instrumented indentation

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is one of the most applied techniques to evaluate the composite properties of the material systems,

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e.g., composite hardness and elastic modulus.

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Hardness analysis from coated materials has been extensively studied in the last decades. This problem has been analyzed from the Bückle’s proposal [5] to more recent researches [6, 7, 8, 9, 10,

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11]. A first attempt to estimate the hardness of a film deposited on a substrate was proposed by

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Bückle [5], who proposes the “10% rule of thumb”. This rule provides that the influence of the substrate on the hardness measurement starts only when the indentation depth, ℎ, reaches at least

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10% of the thickness of the film, 𝑡. However, this is not always true, especially when a soft coating is

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deposited on a hard substrate [12, 13, 14, 15]. Jönsson and Hogmark’s (JH) proposal [6] is one of the first models advanced to characterize small-volume materials (e.g. thin coatings) from indentation tests. They present a simple model based on the mechanical response of the coating during indentation measurements. The composite hardness (𝐻𝑐 ) is expressed by a weighted function of the film and substrate hardness (𝐻𝑓 and 𝐻𝑠 , respectively), i.e., 𝐻𝑐 = 𝛼𝑓 ∙ 𝐻𝑓 + 𝛼𝑠 ∙ 𝐻𝑠 . In their work, the contribution of film and substrate hardness (𝛼𝑓 and 𝛼𝑠 , respectively) are given by the film and substrate areas that react to the contact pressure. This contribution is based on the deformation mode (i.e. plastic or brittle) during indentation, and the geometry of the indenter [16]. The sum of the factors related to the film and substrate hardness is 𝛼𝑓 + 𝛼𝑠 = 1. Thus, the contribution factor 𝛼𝑓 stands for the contact area fraction of the film to the total contact area reacting to the contact pressure, it is expressed as 𝛼𝑓 = 2 ∙

2

𝐶∙𝑡 ℎ

−

𝐶 2 ∙𝑡 2 ℎ2

[6, 16]. When coating deforms

Journal Pre-proof plastically following the shape of the Vickers indenter the constant is 𝐶1 = sin2 (22°). On the other hand, when a brittle coating fractures forming cracks in the interior of the indenter impression the constant is 𝐶2 = 2 sin2 (11°) [6]. These constants can be also established for Berkovich, conical and cube corner indenters [16]. Nevertheless, when the JH model is applied to a coated system and the product 𝐶 ∙ 𝑡 is less than the indentation depth, the contribution factor is negative. These authors infer themselves that their model is not suitable when the indentation depth is less than the thickness of the coating, which resulted in many criticisms in the literature [17, 18, 19]. To improve

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the understanding and the interpretation of the model, Iost and Bigot [20] proposed that the

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it means that composite hardness is the film hardness.

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contribution of the film hardness to the composite hardness should be equal to unity when 𝐶 ∙ 𝑡 < ℎ,

In the recent past years, multilayered systems or graded coatings have been developed to improve

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the mechanical response of material systems [21, 22, 23]. Thus, the composite hardness of this new

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architecture of multilayer coating systems considers the contribution of all components (n layers) of the coating. Recently, many works have been advanced to solve the question of the contribution of a

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single layer to the composite hardness [9, 16, 24, 25]. Puchi-Cabrera et al. [24] developed a

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multilayer model based on the work proposed by Rahmoun et al. [16] where the single-layer Jonssön and Hokmark’s model [6] is generalized to multilayered coatings.

It is important to discriminate the hardness obtained from measurements on the cross-section of the sample (Figure 1 (a)) and the composite hardness corresponding to normal indentation tests on the surface sample (Figure 1 (b)). Figure 1 (a) illustrates the problem treated by Jönsson and Hogmark [6] for a single film deposited on a substrate, both with constant hardness. Figure 1 (a) shows the hardness profile of the coating as a function of the distance from the surface (𝐷) when the indentation is performed on the cross-section of the specimen (𝐻(𝐷𝑖 ). Figure 1 (b) shows the composite hardness (𝐻𝑐 (ℎ𝑖 )) obtained when indentation is performed at the normal surface of the coated system. In this example, hardness of the 50 μm thick coating and of the substrate are

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Journal Pre-proof respectively of 2.5 GPa and 1.0 GPa, and the film is plastically strained to match the shape of the Vickers diamond tip. It is worthy to observe that composite hardness describes the “resultant” hardness from the contribution of the film and substrate hardness. On the other hand, the hardness profile in the cross-section of the specimen is independent of the indentation depth (disregarding the indentation size effect [20]), and probably differences could be observed as a function of the distance from the surface edge of the coated sample if there is a hardness gradient, for example, when the specimen has been subjected to thermochemical or heat treatments [26, 27]. Indeed, some of the

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last proposed multilayered models [16, 24] rely on the hypotheses that the hardness of coatings

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remains constant along the thickness, it means, the hardness is a discontinuous piecewise-defined

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function (Figure 1 (a)).

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----------------------------------Figure 1: (a) Hardness profile determined from cross-sectional indentation tests, and (b) hardness profile determined from normal indentation tests. ------------------------------------

The aim of this work is to evaluate the hardness profile, as a function of the distance to the surface,

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from the composite hardness obtained by hardness tests carried out on the surface of the material

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with increasing loads. Thus, the advantage of defining the composite hardness by a continuousdefined function is that the integral formula proposed here permits to analyze Functionally Graded Materials (FGMs) predicting the hardness profile without any destructive and time-consuming process. To connect our proposition and the historical work of Jönsson and Hogmark [6], it will be demonstrated that their model is a particular case of the formulation proposed in the present work.

2

The model: Towards an integral formula

As explained above, the JH proposal [6] is a geometrical model where the composite hardness is expressed as a linear law of mixture in terms of the film and substrate hardness. These authors considered that for a film of thickness 𝑡, the contribution factor of the film hardness to the composite hardness based on the contact area fraction was proportional to 𝐶 ∙ 𝑡, with 𝐶 a constant

4

Journal Pre-proof less than 1, related to the mode of the film deformation. So, the coefficient 𝐶 depends of the contact’s physics. This reasoning leads to discriminate the definition of hardness, which is expressed as a pressure (perpendicular force per unit area), and the concept of the mechanical contact pressure exerted on the indenter. To have a clear physical meaning, a new function will be introduced. The function 𝛱(𝑝) is the contact pressure exerted on a section of the indenter by the material at a normal distance 𝑝 from the top surface (Figure 2), thus, 𝑝 is a variable which lies in the interval [0, ℎ]. When the hardness is constant

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(𝐻0 ), it means that the mechanical pressure 𝛱(𝑝) = 𝐻0 , ∀ 𝑝 ∈ [0, ℎ]. To illustrate the physical

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meaning of 𝛱(𝑝), it will be considered that the pressure is applied on the projected area (commonly

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used in nanoindentation analysis), and not on the surface area as usual when Vickers Hardness is computed from the indentation length, 𝑑. Considering that the pyramidal indenter is sliced in a

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perpendicular direction of the principal axe, the differential projected area is: 𝑑𝐴𝑝 = 8𝑥𝑑𝑥 (Figure

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2).

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---------------------------------------------Figure 2: Representation of the function parameters used for a Vickers indenter. ----------------------------------------------

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On one hand, the equivalent force from the composite hardness is given by the product of the total exerted contact pressure (composite hardness) and the square area from the indentation length: 𝑑 2√2

𝑑 2√2

𝐿=∫

𝐻c (ℎ)8𝑥𝑑𝑥 = 8𝐻c (ℎ) ∫

0

0

𝑑2 𝑥𝑑𝑥 = 𝐻 (ℎ) 2 c

(1)

On the other hand, the sum of the contribution of the contact pressure exerted on the projected 𝑑

areas (from 𝑥 = 0 to 𝑥 = 2 2) is written as follows: √

𝑑 2√2

𝑑 2√2

𝐿=∫

𝛱(𝑝)8𝑥𝑑𝑥 = 8 ∫

0

0

(2)

𝑥𝛱(𝑝)𝑑𝑥

The equality between forces in equations (1) and (2), gives us: 𝑑

𝑑2 2√2 𝐻𝑐 (ℎ) = 8 ∫ 𝑥𝛱(𝑝)𝑑𝑥 2 0

(3)

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Journal Pre-proof From Figure 2, it is possible to establish the relationship 𝑑

substitution of 𝑝 = (2 𝐻𝑐 (ℎ) =

√2

ℎ 𝑑 2√2

=

𝑝 (

𝑑 −𝑥) 2√2

= tan 𝛼, so with the

− 𝑥 )tan 𝛼 which involves 𝑑𝑝 = − tan 𝛼 𝑑𝑥, it is found:

2 ℎ ∫ (ℎ − 𝑝)𝛱(𝑝) 𝑑𝑝 ℎ2 0

(4)

This formula gives the relationship between the normal hardness (or composite hardness, 𝐻𝑐 (ℎ)) and the pressure exerted on an indenter section by a homogeneous and isotropic material (function 𝛱(𝑝)) and is the milestone of the present work.

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Switching from pressure contact, 𝚷(𝒑), to hardness profile, 𝑯(𝑫)

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according to the distance to the surface 𝐷 will be given.

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In a second time, the relationship between the function 𝛱(𝑝) and the cross-section hardness 𝐻(𝐷)

It is important to note that 𝐻c (ℎ) stands for the composite hardness at the indentation depth ℎ

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(Figure 1 (b)), and 𝐻(𝐷) denotes the hardness of a cross-section located at a distance 𝐷 from the

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upper surface (Figure 1 (a)). In the case of film hardness of a monolayer-coated system, Jönsson and

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Hogmark [6] applied a simple mathematical method which consists in a dilatation:

(5)

𝐻(𝐷) = 𝛱(𝑝), with 𝑝 = 𝐶 ∙ 𝐷

Eq. (5) means that the contact pressure exerted on one normal section of the indenter at a distance 𝑝 = 𝐶 ∙ 𝐷 from the top surface equals the cross-section hardness at the distance 𝐷 from the top surface. In this work, the constant 𝐶 is defined according the deformation modes proposed by Jönsson and Hogmark [6].

4

Monolayer model from the integral approach

At this point, it is useful to make the link between Eq. (4) and the work of Jönsson and Hogmark [6]. From a single film on a substrate and retaining the assumption that in a thickness 𝐶 ∙ 𝑡 the action on

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Journal Pre-proof the indenter is exerted only by the film, the contact pressure exerted on a normal section of the indenter at a distance 𝑝 from the top surface is represented in Figure 3. In this case, 𝛱(𝑝) is a piecewise-defined discontinuous function, where 𝐻𝑓 and 𝐻𝑠 denote the contact pressure applied on the indenter by the film and substrate, respectively, as follows: 𝐻𝑓 0≤𝑝 ≤𝐶∙𝑡 if 𝛱(𝑝) = { 𝐻𝑠 otherwise

(6)

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-----------------------------------Figure 3: Representation of the contact pressure exerted on a normal section of the indenter at a distance 𝑝 from top surface. ------------------------------------

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From Eq. (4), it is possible to calculate 𝐻c (ℎ). From Eq. (6) and for 0 ≤ ℎ ≤ 𝐶 ∙ 𝑡, we have 𝛱(𝑝) = 𝐻𝑓 ,

2 ℎ ∫ (ℎ − 𝑝)𝐻𝑓 𝑑𝑝 = 𝐻𝑓 ℎ2 0

(7)

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𝐻𝑐 1 (ℎ) =

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and then from Eq. 4:

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From Eq. (6) and for ℎ > 𝐶 ∙ 𝑡, as 𝛱(𝑝) contains two pieces, and taking into account that the contribution of the substrate to the composite hardness starts from 𝑝 = 𝐶 ∙ 𝑡, it is written: 𝐶∙𝑡 ℎ 2 ℎ 2 (ℎ (ℎ (ℎ − 𝑝)𝐻𝑠 𝑑𝑝) ∫ − 𝑝)𝛱(𝑝) 𝑑𝑝 = − 𝑝)𝐻 𝑑𝑝 + ∫ (∫ 𝑓 ℎ2 0 ℎ2 0 𝐶∙𝑡

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𝐻𝑐 2 (ℎ) =

𝐻𝑐 (ℎ) = 𝐻𝑓 (2

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(𝐶 ∙ 𝑡)2 (𝐶. 𝑡)2 2 ℎ2 2 𝐻𝑐 (ℎ) = 2 (𝐻𝑓 (ℎ(𝐶 ∙ 𝑡) − ) + 𝐻𝑠 ((ℎ − ) − (ℎ(𝐶. 𝑡) − ))) ℎ 2 2 2

(8)

𝐶 ∙ 𝑡 𝐶2 ∙ 𝑡2 𝐶 ∙ 𝑡 𝐶2 ∙ 𝑡2 − + 𝐻 − 2 + ) (1 ) 𝑠 ℎ ℎ2 ℎ ℎ2

The whole solution has two continuous pieces in 𝐶 ∙ 𝑡 written as follows: 𝐻𝑐 (ℎ) = {

𝐻𝑐 1 𝐻𝑐 2

if 0≤ℎ ≤𝐶∙𝑡 otherwise

(9)

This result (Eq. (9)) corresponds to the JH model modified by Iost and Bigot [20] for indentation depths lower than the thickness of the coating multiplied by the coefficient 𝐶. Iost and Bigot’s proposal [20] appears to be mathematically inevitable for a suitable treatment of Eq. (4).

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Journal Pre-proof 5

Multilayer model from integral approach -------------------------------Figure 4: Representation of the contact pressure exerted on a normal section of the indenter by a 𝑗 multilayer-coating and substrate system at a distance 𝑝 from top surface, where 𝑝𝑗 = ∑𝑖=1 𝐶𝑖 ∙ 𝑡𝑖 --------------------------------

In this section, the same reasoning will be used to deduce a multilayer model from the integral approach. Suppose a multilayered coating having 𝑛 layers with thicknesses 𝑡𝑖 (Figure 4) with 𝑖 = 1, ⋯ , 𝑛. For each layer, the thickness 𝑡𝑖 is associated to a coefficient 𝐶𝑖 and to a constant

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hardness 𝐻𝑖 . The substrate is also assumed to have a constant hardness 𝐻𝑠 . It will be supposed that

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the indentation depth is such that ℎ > ∑𝑛𝑖=1 𝐶𝑖 ∙ 𝑡𝑖 . For the first piecewise function of 𝐻𝑐 (ℎ) (when

2 ℎ ∫ (ℎ − 𝑝)𝛱(𝑝) 𝑑𝑝 ℎ2 0

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𝐻𝑐 𝑛 (ℎ) =

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0 ≤ ℎ ≤ 𝐶 ∙ 𝑡), Eq. (4) gives 𝐻𝑐 1 (ℎ) = 𝐻1 , and for the last piecewise function (ℎ > ∑𝑛𝑖=1 𝐶𝑖 ∙ 𝑡𝑖 ):

𝑖

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∑𝑗=1 𝐶𝑗 ∙𝑡𝑗 𝐶1 ∙𝑡1 2 (ℎ − 𝑝)𝐻1 𝑑𝑝 + ⋯ + ∫ (ℎ − 𝑝)𝐻𝑖 𝑑𝑝 + ⋯ = 2 (∫ ℎ ∑𝑖−1 𝐶𝑗 ∙𝑡𝑗 0 ℎ ∑𝑛 𝑗=1 𝐶𝑗 ∙𝑡𝑗

(ℎ − 𝑝)𝐻𝑠 𝑑𝑝)

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+∫

2

∑𝑖

𝐶𝑗 ∙𝑡𝑗

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𝑗=1 The general integral term 𝐼𝑖 = ℎ2 ∫∑𝑖−1

𝑗=1 𝐶𝑗 ∙𝑡𝑗

𝑖

(10)

𝑗=1

2

(ℎ − 𝑝)𝐻𝑖 𝑑𝑝 is: 2

𝑖−1

(∑𝑖𝑗=1 𝐶𝑗 ∙ 𝑡𝑗 ) (∑𝑖−1 2𝐻𝑖 𝑗=1 𝐶𝑗 ∙ 𝑡𝑗 ) 𝐼𝑖 = 2 ((ℎ ∑ 𝐶𝑗 ∙ 𝑡𝑗 − ) − (ℎ ∑ 𝐶𝑗 ∙ 𝑡𝑗 − )) ℎ 2 2 𝑗=1

𝑗=1

(11)

2

2

(∑𝑖𝑗=1 𝐶𝑗 ∙ 𝑡𝑗 ) (∑𝑖−1 𝑗=1 𝐶𝑗 ∙ 𝑡𝑗 ) 𝐼𝑖 = 𝐻𝑖 ((1 − (1 − ) ) − (1 − (1 − ) )) ℎ ℎ In fact, the whole expression is obvious to calculate and is identical to that of the Puchi-Cabrera et al.’s [24] model.

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Journal Pre-proof 6

Numerical approximation of the integral approach by the multilayer formula

6.1

Numerical application

The articulation between the multilayer model and the integral approach is illustrated through an example based on a pressure gradient section (Figure 5) between two constant pressure sections. In analogy with the numerical approximation of the rectangle method, it is possible to observe that creating 6 layers (Figure 5(a)) to replace the gradient region, the pressure of each layer could be defined as follows: 𝑖 = 1, from 0 to 5 µm, a pressure of 5 GPa,



𝑖 = 2, from 5 to 9 µm, a pressure of 4.40 GPa,



𝑖 = 3, from 9 to 13 µm, a pressure of 3.80 GPa,



𝑖 = 4, from 13 to 17 µm, a pressure of 3.20 GPa,



𝑖 = 5, from 17 to 21 µm, a pressure of 2.60 GPa,



𝑖 = 6, from 21 to 25 µm, a pressure of 2 GPa; after, the substrate is reached.

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

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---------------------------------Figure 5: (a) The gradient section of the function 𝛱(𝑝) is approximated with 6 layers, (b) comparison between the composite hardness, 𝐻𝑐 (ℎ), obtained by the multilayer approach (6 layers) and the continuous linear gradient modeling of 𝛱(𝑝). ---------------------------------Now, to consider the integral approach, a pressure function 𝛱(𝑝) will be defined. In this case, 𝛱(𝑝) is a piecewise-defined continuous function modelling the contact pressure exerted on a normal section of the indenter at a distance 𝑝 from the top surface, where 𝐻𝑓 and 𝐻𝑠 denote the film and substrate contact pressure, respectively, and 𝛱2 (𝑝) is a linear function defining a pressure gradient. It can be written as follows: 𝐻𝑓 𝛱(𝑝) = {𝛱2 (𝑝) 𝐻𝑠

0 ≤ 𝑝 ≤ 𝑝1 if 𝑝1 < 𝑝 ≤ 𝑝2 if otherwise

(12)

In Figure 5(a), the function 𝛱(𝑝) corresponds to a 25 µm thick film, with a constant hardness of 5 GPa over the first 5 µm (𝐻𝑓 ), and a pressure gradient with a linear behaviour 𝛱2 (𝑝) at distances 9

Journal Pre-proof between 5 µm and 25 µm from the top surface. The substrate has a constant hardness of 2 GPa (𝐻𝑠 ). Thus, this function can be expressed according to Eq. (12) as: 5 𝛱(𝑝) = {5.75 − 0.15𝑝 2

0≤𝑝≤5 if 5 < 𝑝 ≤ 25 if otherwise

(13)

The Figure 5(b) allows us to observe the differences between considering a multilayer model defined from a discontinuous piecewise-defined function and the integral approach by a continuous linear

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pressure gradient. By refining the partition of the gradient interval, a good concordance between the

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multilayer formula and the integral approach was obvious and it is improved with a larger number of

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layers or by using the trapezoidal numerical method of integration.

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In Figure 5(b) the continuous curve represents the true and continuous behavior of the composite

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as 𝛱(𝑝).

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hardness 𝐻𝑐 (ℎ) when the integral approach is implemented. The response 𝐻𝑐 (ℎ) has the same unit

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The difficulty in using the hardness gradient model is to identify the continuous model for parametric decomposition. A possible solution is to propose a two steps method: first, a multilayer analysis to approximate the continuous model, and then, the integral formula to perform a parametric analysis with fewer parameters. In practice, there are 𝑛 experimental ordered pairs (ℎ𝑖 , 𝐿𝑖 ), where ℎ𝑖 is the indentation depth and 𝐿𝑖 the applied load on the indenter from the load-displacement curve. Then, 𝑛 ordered pairs (ℎ𝑖 , 𝐻𝑖 ) are computed where 𝐻𝑖 = 𝐻𝑐 (ℎ𝑖 ) is the composite hardness. At this point the function 𝛱(𝑝) is searched. This is a typical inverse problem. In our examples, the hardness from the distance to the surface (𝐻(𝐷)) or the function 𝛱(𝑝) is never used to calculate the function 𝐻𝑐 (ℎ). It is only used to compare the model with experimental data.

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Journal Pre-proof 6.2

Theory

A parametric function for 𝛱(𝑝) is chosen as a differentiable or piecewise differentiable function noted: 𝛱(𝑝, 𝜃), where 𝜃 stands for the list of adjustment parameters. Thus, Eq. (4) becomes 2

ℎ

𝐻𝑐𝑚𝑜𝑑 (ℎ, 𝜃) = ℎ2 ∫0 (ℎ − 𝑝)𝛱(𝑝, 𝜃) 𝑑𝑝, where 𝐻𝑐𝑚𝑜𝑑 (ℎ, 𝜃) is the modeled composite hardness while 𝑒𝑥𝑝

𝐻𝑐

(ℎ𝑖 ) denotes the experimental values. The residual sum of the squared formula 𝑤(𝜃), the

objective function, is defined as follows to estimate the parameters while searching the minimum of

𝑛

2

ℎ𝑖

𝑒𝑥𝑝 = ∑ ( 2 ∫ (ℎ𝑖 − 𝑝)𝛱(𝑝, 𝜃)𝑑𝑝 − 𝐻𝑐 (ℎ𝑖 )) ℎ𝑖 0 𝑖=1

2

(14)

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∑(𝐻𝑐𝑚𝑜𝑑 (ℎ𝑖 , 𝜃) − 𝑖=1

-p

𝑤(𝜃) =

𝑛 2 𝑒𝑥𝑝 𝐻𝑐 (ℎ𝑖 ))

of

𝑤(𝜃), where the list of optimum parameters is noted 𝜃̂:

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Mathematica software (Wolfram Research) is used to find the optimum parameters. It is well known [28, 29] that with a 4-parameters function, convergence is almost impossible to achieve, with a 3-

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parameters function it is possible with good initial conditions, with a 2-parameters function the

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convergence is rather easy to obtain. In Figure 6, it is natural to suppose that 𝐻1 and 𝐻2 are wellknown values, and 𝑝1 and 𝑝2 are to be calculated, and they stand for the beginning and end,

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respectively, of the graded section on the sample. More precisely, 𝐻1 is the hardness obtained by nanoindentation test performed at the top of the surface, 𝐻2 is the hardness obtained by indentation test on the substrate.

---------------------------------------------------Figure 6: Representation of the contact pressure exerted on a normal section of the indenter by a graded surface and substrate system, with a 4-parameters function (𝐻1 ,𝐻2 , 𝑝1 , 𝑝2 ). ---------------------------------------------------6.3

Numerical stability

It will be considered a 4-parameters gradient function with 𝐻1 = 5.00 GPa, 𝐻2 = 2.00 GPa, 𝑝1 = 5.00 µm and 𝑝2 = 25.00 µm (Figure 5 (a)). In Figure 5 (b) the integral response is sampled with a point per micrometer. It is considered that 𝐻2 = 2.00 GPa is a well-known value, it means, a 3-parameters gradient function. The parameters 𝑝1 , 𝑝2 and 𝐻1 were calculated with Mathematica

11

Journal Pre-proof software assigning good initial conditions [28, 29]. Now, the idea is to study the numerical stability of the results by adding a Gaussian noise to the 𝐻i = 𝐻𝑐 (ℎi ) values of the ordered pairs (ℎi , 𝐻i ). A standard deviation of 0.20 GPa is chosen. For example, a test represented by the curve in (Figure 7) gives 𝐻1 = 4.96 GPa, 𝑝1 = 5.58 µm and 𝑝2 = 24.55 µm . A typical least squares behavior with the curve inside the cloud of points is recognized. ----------------------------------------Figure 7: Effect of noise with 𝐻2 = 2.00 GPa for one Monte Carlo simulation. -----------------------------------------

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The application of Monte Carlo simulation leads to the histograms presented in Figure 8 (a), the

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averages of computed values are 𝐻1 = 4.99 GPa, 𝑝1 = 5.47 µm and 𝑝2 = 24.50 µm.

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-----------------------------Figure 8: Histograms obtained from Monte Carlo simulation for the variation of the surface hardness and the geometric parameters 𝑝 in relation with the hardness of the substrate 𝐻2 . (a) 𝐻2 = 2.00 GPa, (b) 𝐻2 = 2.20 GPa, (c) 𝐻2 = 1.80 GPa --------------------------------

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If it is considered that the hardness of the substrate is not known precisely, it is obtained as averages

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𝐻1 = 4.99 GPa, 𝑝1 = 7.47 µm 𝑝2 = 19.22 µm (Figure 8(b)) and 𝐻1 = 5.02 GPa, 𝑝1 = 3.82 µm, 𝑝2 = 29.48 µm (Figure 8(c)) for substrate hardness 𝐻2 of 2.20 and 1.80 GPa, respectively, and we can see that the histograms become asymmetrical. At this point, it is interesting to evaluate the independence of the variables. For the set of values obtained with 𝐻𝑠 = 2.00 GPa , 𝐻1 is shown in Figure 9 as a function of 𝑝1 and 𝑝2 . There is a clear correlation between these three parameters. The problems of convergence linked to the correlation between the parameters and to the sensibility to the substrate hardness are difficult to avoid when applying computation with multi-parameters modeling functions.

----------------------Figure 9: 3D-Scatter plot showing the correlation between 𝐻1 , 𝑝1 and 𝑝2 ,and their orthogonal projection onto the coordinate planes. ---------------------------------

12

Journal Pre-proof 7 7.1

The parametric modeling function: selection and computation A linear gradient model

To validate the proposed integral approach, the experimental data obtained by Lesage et al. [30] for sample S1 will be used. Lesage et al. [30] proposed a method to evaluate the hardness profile from a series of indentations on the normal surface increasing the applied loads. In their experiments, AISI 4140 steel samples were subjected to a duplex treatment consisting in ion nitriding followed by Zpinch surface irradiation causing variations on the composition from surface to core of the sample

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and consequently a hardness gradient.

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The data from [30], which is used in our work, are indentation tests performed on surface with loads varying from 0.1 to 2500 N and microhardness tests performed on a cross-section of the sample from

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top surface until 500 µm from the top surface. A linear gradient model with 4-parameters

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𝜃 = (𝐻1 , 𝐻2 , 𝑝1 , 𝑝2 ) is used (as in Figure 6). The hardness of the substrate being supposed to be 𝐻𝑠 = 2.50 GPa, one will calculate the optimum values of others three parameters. Searching the

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minimum of Eq.(14) and considering a plastic deformation of the film (𝐶 = sin2 (22°)), it is found,

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𝐻𝑓 = 8.07 GPa, 𝑝1 = 0 µm and 𝑝2 = 21.88 µm. Figure 10 (a) shows the agreement between the curve defined by Eq. (4) and the experimental data (bold squares) and Figure 10 (b) gives the hardness as a function of the distance from the top surface, marks are experimental results and the continuous line is the function 𝐻(𝐷) = 𝛱(𝑝) with 𝑝 = 𝐶 ∙ 𝐷. Thus, 𝐷1 =

𝑝1 𝐶

and 𝐷2 =

𝑝2 −𝑝1 . 𝐶

In Figure

10, bold squares represent the experimental values of [30]. The curve in Figure 10 (a) is the result of the Eq. (4) considering a pressure gradient function, 𝛱(𝑝), after it has been identified as shown in Figure 10 (b) by minimizing the Eq. (14). Both functions, 𝐻𝑐 (ℎ) and 𝐻(𝐷), represent very well the experimental data from [30].

13

Journal Pre-proof ------------------------------Figure 10: (a) Composite hardness as a function of indentation depth, 𝐻𝑐 (ℎ), performed by indentation on the top of the specimen with loads from 0.1 to 2500 N, (b) Hardness as a function of the distance from the top surface, 𝐻(𝐷), obtained by microindentation tests performed on the crosssection of the sample. For the two figures, the bold squares correspond to experimental data performed by Lesage et al. [30]. ------------------------------7.2

A sigmoid function: The Gauss error function

The Fick’s second law describes the concentration distribution of chemical species as a function of a distance 𝑥 and the diffusing time 𝜏. Recent research describing the microhardness of steel after a

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based on the solution of the Fick’s second law as follows:

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nitriding process [31] proposed to evaluate the hardness as a function of the nitriding propagation

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𝑐 − 𝑐0 𝑥 = 1 − erf( ) 𝑐𝑠 − 𝑐0 2√𝐷𝑓 𝜏

(15)

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The concentration distribution is expressed in Eq. (15), where 𝑐 is the concentration of the chemical

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specie at distance 𝑥 after time 𝜏; with boundary conditions: for 𝜏 = 0, 𝑐 = 𝑐0 and for 𝜏 > 0, 𝑐 = 𝑐𝑠 at 𝑥 = 0 and 𝑐 = 𝑐0 at 𝑥 = ∞. 𝐷𝑓 is the diffusion coefficient and the notation erf indicates the

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Gaussian error function. Thermochemical treatments (carburizing, nitriding, carbonitriding …) usually

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span from depths of several tens of micrometers and the mechanical properties vary as a function of the penetration of the chemical specie. From the fact that the microstructure of the material, and consequently the mechanical properties change as a function of the distance from the top surface after a thermochemical or heat treatment, a hardness profile following a sigmoid gradient model is proposed: 𝑝+𝑏 𝑝+𝑏 𝛱(𝑝) = (1 − erf ( )) ∙ 𝐻1 + erf ( ) ∙ 𝐻𝑠 𝑎 𝑎

(16)

It is observed that the limit of 𝛱(𝑝) is 𝐻𝑠 when 𝑝 tends towards infinite, with 𝑏 = 0, and 𝛱(𝑝 = 0) = 𝐻1. Eq. (16) will allow us to predict the hardness profile on the cross-section of the sample.

14

Journal Pre-proof The Figure 11 shows the same experimental data from [30] than in Figure 10. The values 𝐻𝑠 = 2.50 GPa and 𝑏 = 0 were set. It was found, from the sigmoid gradient model expressed in Eq. (16), 𝐻1 = 8.17 GPa and 𝑎 = 20.20 µm. This data from [30] was obtained from a sample with a surface treated through ion nitriding. Regarding the nature of the diffusion phenomenon of the thermochemical surface treatment, Eq. (16) provides a better fitting of the behavior of normal composite hardness and cross-section hardness profile of the sample than simply considering a linear gradient model as presented in Figure 10.

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-------------------------------------Figure 11: (a) Normal hardness obtained by indentation tests on the normal surface [30] (bold squares), 𝐻𝑐 (ℎ), modeled by an erf sigmoid function (continuous curve), (b) erf sigmoid function (continuous curve) proposed to model the hardness profile 𝐻(𝐷) of the experimental data (bold squares) [30]. --------------------------------------

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Other results available in the literature were also tested, the Influence of the combined processes of

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plasma nitriding and cathodic arc ion plating on high-speed steel substrates by Ichimura et al. [32]; and the effect of plastic deformation surface layer by means of surface mechanical attrition

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treatment before low-temperature nitriding of an AISI 321 stainless steel with nanocrystalline grains

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from Lin et al. [33]. For the two previous researches, experimental data of normal composite hardness variation with indentation depth and hardness profile as a function of the distance from the top surface are available. The model presented here was applied for hardness tests carried out on the surface using increasing loads and validated for hardness profiles on a cross-section of the sample with a very good accuracy. The hardness profile is computed from normal hardness measurements performed with increasing loads. Then, the 𝛱(𝑝) is defined as a function of the nature of the system, e.g. Eqs. (12) or (16), and from Eq. (14), the parameters are estimated by searching the minimum of the objective function ( 𝑤(𝜃)) from Eq. (14). Worth to point out that the diffusion phenomenon supposes a homogeneous form of hardening, the solid solution one more likely, where the approach suits to be applied. Nonetheless, if there is a

15

Journal Pre-proof precipitation hardening the model will not be longer valid, and a different approach should be considered by evaluating the hardness of the heterogeneous materials [34]. Nevertheless, It should be noted that, in this approach, it was used as initial data the normal composite hardness variations of samples obtained by increasing applied loads; these data may need to be corrected taking into account the phenomenon of indentation size effect (ISE) when this last is important [20, 35, 36, 37]. Considering the applied load range used to perform the developed approach, the ISE would be related to the plastic deformation around the indenter and the

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consequent pile-up formation around the indentation imprint, and not to the strain gradient

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plasticity theory [38, 39], which is based on the geometrically necessary dislocations, applied at very

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shallow depths. Moreover, researches have shown the relationship between the pile-up displacements and the stress-state of the system [40, 41]. So, since the ISE and the residual stresses

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produce a similar mechanical response, a first approach from [16, 24] for the outer layer of a

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multilayered system could consist in the definition of the function 𝛱(𝑝) = 𝐻𝑓0 + 𝑓(𝑝) when

defined as 𝑓(𝑝) =

𝛽𝑓 √𝑝

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0 ≤ 𝑝 ≤ 𝐶 ∙ 𝑡, where 𝐻𝑓0 is the independent-of-ISE hardness of the outer film, and 𝑓(𝑝) would be under the strain gradient plasticity theory [38, 42] or as 𝑓(𝑝) =

𝛾𝑓 𝑝

under the

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effect of the pile-up development in the vicinity of the indenter [43, 44]. But this implies the addition of one adjustment parameter (𝛽𝑓 or 𝛾𝑓 ) in the developed model that could affect the robustness of the approach above described. Considering, on one hand, that this problem is encountered at the surface of the film and, on the other hand, that one principal application of the proposed approach is the prediction of the hardness profile of mechanical parts subjected to thermochemical treatments which involves indentation tests at a high load range, in the present model, the ISE is considered to be neglectable, but a better understanding on modelling this issue must be developed when applying it in thin films. Bearing in mind these two issues, the ISE and the residual stresses, when performing indentation on the normal surface, the hardness measurements would be affected by stress-state into the layers

16

Journal Pre-proof and by the residual stresses at layer’s interfaces on multilayered systems. Thus, since a high load range would be used to perform these hardness measurements, the contribution of the ISE to the pile-up formation would be much smaller than the contribution of the residual stresses, so the ISE could be neglected for hardness calculation. Conversely, when performing indentation on the crosssectional surface, a very low load range is applied, and the contribution of the ISE may be important. Accordingly, it would be possible to find hardness differences between normal and cross-sectional measurements. On one hand, this can be explained by the differences in applied load ranges at

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normal and cross-sectional measurements and the contribution of the indentation size effect.

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On the other hand, the residual stresses would influence the hardness measurement performed at

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the surface of the material. This influence would be a mean of the gradient of residual stresses in a

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representative zone under the indenter tip and it would not be representative of one at a given distance from the top surface of the specimen. Hardness measurement performed on a cross-section

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of the material should be more sensitive to residual stresses, but unfortunately these stresses are

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partly relaxed by the fact that the sample is cut, and their effect is hidden by that of the ISE. Therefore, another contribution of this innovative approach would be that the hardness profile

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predicted from the developed model is less dependent on the indentation size effect and on the residual stresses than direct measurements performed on a cross section of the specimen. 8

Searching the modeling function directly from the integral approach

Previously, the composite hardness 𝐻𝑐 (ℎ) was well-known as experimental data from indentation tests, and from numerical analysis using the integral approach (Eq. (4)) the function 𝛱(𝑝) was deduced. Now, a parametric function of 𝐻𝑐 (ℎ) is chosen to be at least a second-order differentiable function of Eq. (4), 𝐻𝑐𝑚𝑜𝑑 (ℎ, 𝜃) to find 𝛱(𝑝). The minimization function is set to be 𝑒𝑥𝑝

𝑤(𝜃) = ∑𝑛𝑖=1(𝐻𝑐𝑚𝑜𝑑 (ℎ𝑖 , 𝜃) − 𝐻𝑐

2 (ℎ𝑖 )) to estimate the optimal parameters 𝜃̂. To simplify the

notation, 𝐻𝑐𝑚𝑜𝑑 (ℎ, 𝜃) will be designated as 𝐻𝑐 (ℎ). In this form, a direct decomposition is possible. The second derivative of Eq. (4) gives:

17

Journal Pre-proof 1 𝛱(ℎ) = 𝐻𝑐 (ℎ) + 2ℎ ∙ 𝐻𝑐′ (ℎ) + ∙ ℎ2 ∙ 𝐻𝑐′′ (ℎ) 2

(17)

Considering Eq. (17) as a differential equation, 𝐻𝑐 (ℎ) could be called a particular solution. To find the general solution, the homogeneous differential equation is written: 1 𝐻𝑐 (ℎ) + 2ℎ𝐻𝑐′ (ℎ) + ℎ2 𝐻𝑐′′ (ℎ) = 0 2

(18)

If Eq. (18) has a solution, this solution gives a complete solution for 𝛱(ℎ) in addition with a particular solution.

To

define

the

complete

solution,

the

boundary

conditions

are

known:

−2𝐶𝑡(𝐻𝑠 −𝐻𝑓 ) ℎ

+

𝐶 2 𝑡 2 (𝐻𝑠 −𝐻𝑓 ) ℎ2

is found to be the unique solution verifying

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and Hogmark’s solution: 𝐻𝑠 +

of

(i) limℎ→0 𝛱(ℎ) = 𝐻𝑓 , (ii) limℎ→+∞ 𝛱(ℎ) = 𝐻𝑠 , and (iii) 𝛱′(ℎ = 0) = 0 (see appendix A). The Jönsson

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the two continuity conditions C0 and C1 in the general solution. From an analogous reasoning, the ∑𝑖𝑗=2 −2∙𝑝𝑗−1 ∙(𝐻𝑗 −𝐻𝑗−1 )

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model for multilayer coatings was found to be: 𝐻𝑖 + 𝑗

ℎ

+

2 ∑𝑖𝑗=2 𝑝𝑗−1 ∙(𝐻𝑗 −𝐻𝑗−1 )

ℎ2

with

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[24] after arranging the terms.

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𝑝𝑗 = ∑𝑖=1 𝐶𝑖 ∙ 𝑡𝑖 for the 𝑖 th-layer (𝑖 = 2, ⋯ , 𝑛), being the solution proposed by Puchi-Cabrera et al.

Thus, Eq. (17) gives the function 𝛱(𝑝) if 𝐻𝑐 (ℎ) is a well-known function, or 𝐻𝑐 (ℎ) if 𝛱(𝑝) is a known

interesting. 9

Conclusion

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function by solving the differential equation. For the latest case, the integral form is more

The integral formula (Eq. (4)) generalizes the method of Jönsson and Hogmark [6] as well as the multilayer methods [16, 24] and is consistent with the modification proposed by Iost and Bigot [20]. The mathematical treatment of the equation is described in detail with a scheme from 𝐻𝑐 (ℎ) to 𝛱(𝑝) by an integral approach (Eq. (4)), and a path from 𝐻𝑐 (ℎ) to 𝛱(𝑝) with a differential equation (Eq.(17)). The parametric approximation is typically an inverse problem. The method based on the integral formula makes possible to analyse the normal composite hardness variations due to diffusion phenomena in thermochemical surface treatments (nitriding, carburizing, carbonitriding or nitrocarburizing) to predict the hardness profile on the cross-section of the mechanical part, and 18

Journal Pre-proof then to control the carburising- or nitriding- depth avoiding destructive and consuming-time methods. The method proposed here has been developed for indentation tests available in the literature that were performed with Vickers diamond indenters, but depending on the thickness of the graded surface (or the multilayer coating) and on the hardness of different phases, the method can be easily extrapolated to any kind of indenter, especially Berkovich or cubic corner indenters used in nanoindentation and applications as very thin graded surfaces used overall in electronics, optical, sol-gel applications, etc. It is also possible to apply this method to estimate the Young's

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modulus variation as a function of the distance to the surface by means of normal instrumented

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indentation tests, since the corresponding model has already been developed and used for

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multilayer materials [34, 45].

The presence of the ISE and residual stresses and their impact on hardness are a current limitation of

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the model proposed here, so practitioners should be sure that ISE or residual stress issues can be

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neglected. A long-term goal is to develop works addressing the stretching component of films and how that would be impacted by the ISE and residual stresses. Nevertheless, since normal

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measurements of hardness are influenced by the stress-state of the sample, the profile hardness

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deduced from the advanced approach would be less sensitive to the indentation size effect and to the relaxation of residual stresses occurring by sectioning the sample before indentation.

Acknowledgements:

This work was financially supported by funds from the European Community Program InterReg V “TRANSPORT” (Program France-Wallonie-Vlaanderen).

19

Journal Pre-proof References [1] ASTM_Standard_E92-17, Standard Test Methods for Vickers Hardness and Knoop Hardness of Metallic Materials, West Conshohocken, PA: ASTM International, 2017. [2] ASTM_Standard_E111-17, Standard Test Method for Young’s Modulus, Tangent Modulus, and Chord Modulus, West Conshohocken, PA: ASTM International, 2017. [3] ASTM_Standard_E1820-18, Standard Test Method for Measurement of Fracture Toughness,

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West Conshohocken, PA: ASTM International, 2018.

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[4] ASTM_Standard_E8/E8M-16a, Standard Test Methods for Tension Testing of Metallic Materials,

-p

West Conshohocken, PA: ASTM International, 2016.

lP

Techniques du Ministère de l'Air, 1960.

re

[5] H. Bückle, L'essai de microdureté et ses applications, vol. 90, Paris: Publications Scientifiques et

no. 3, pp. 257-269, 1984.

na

[6] B. Jönsson and S. Hogmark, "Hardness measurements of thin films," Thin Solid Films, vol. 114,

Jo ur

[7] P. J. Burnett and D. S. Rickerby, "The mechanical properties of wear-resistant coatings: I: Modelling of hardness behaviour," Thin Solid Films, vol. 148, no. 1, pp. 41-50, 1987. [8] D. Chicot and J. Lesage, "Absolute hardness of films and coatings," Thin Solid Films, vol. 254, no. 1-2, pp. 123-130, 1995. [9] D. S. Stone, "Elastic rebound between an indenter and a layered specimen: Part I. Model," Journal of Materials Research, vol. 13, no. 11, pp. 3207-3213, 1998. [10] S. J. Bull, "Modelling the hardness response of bulk materials, single and multilayer coatings," Thin Solid Films, vol. 398, pp. 398291-298, 2001.

20

Journal Pre-proof [11] E. S. Puchi-Cabrera, "A new model for the computation of the composite hardness of coated systems," Surface and Coatings Technology, vol. 160, no. 2-3, pp. 177-186, 2002. [12] K. Yoder, D. Stone, R. Hoffman and J. Lin, "Elastic rebound between an indenter and a layered specimen: Part II. Using contact stiffness to help ensure reliability of nanoindentation measurements," Journal of Materials Research, vol. 13, no. 11, pp. 3214-3220, 1998. [13] R. Saha and W. Nix, "Effects of the substrate on the determination of thin film mechanical

of

properties by nanoindentation," Acta Materialia, vol. 50, no. 1, pp. 23-38, 2002.

ro

[14] Y.-C. Huang and S.-Y. Chang, "Substrate effect on mechanical characterizations of aluminum-

-p

doped zinc oxide transparent conducting films," Surface and Coatings Technology, vol. 204, no.

re

20, pp. 3147-3153, 2010.

lP

[15] N. Moharrami and S. Bull, "A comparison of nanoindentation pile-up in bulk materials and thin films," Thin Solid Films, vol. 572, pp. 189-199, 2014.

na

[16] K. Rahmoun, A. Iost, V. Keryvin, G. Guillemot and N. C. Sari, "A multilayer model for describing

Jo ur

hardness variations of aged porous silicon low-dielectric-constant thin films," Thin Solid Films, vol. 518, no. 1, pp. 213-221, 2009.

[17] A. M. Korsunsky, M. R. McGurk, S. J. Bull and T. F. Page, "On the hardness of coated systems," Surface and Coatings Technology, vol. 99, no. 1-2, pp. 171-183, 1998. [18] D. Beegan, S. Chowdhury and M. T. Laugier, "Modification of composite hardness models to incorporate indentation size effects in thin films," Thin Solid Films, vol. 516, no. 12, pp. 38133817, 2008. [19] G. Guillemot, A. Iost and D. Chicot, "Comments on the paper “Modification of composite hardness models to incorporate indentation size effects in thin films”, D. Beegan, S. Chowdhury

21

Journal Pre-proof and MT Laugier, Thin Solid Films 516 (2008), 3813–3817," Thin Solid films, vol. 518, no. 8, pp. 2097-2101, 2010. [20] A. Iost and R. Bigot, "Hardness of coatings," Surface and Coatings Technology, vol. 80, no. 1-2, pp. 117-120, 1996. [21] C. Tritremmel, R. Daniel, H. Rudigier, P. Polcik and C. Mitterer, "Mechanical and tribological properties of AlTiN/AlCrBN multilayer films synthesized by cathodic arc evaporation," Surface

of

and Coatings Technology, vol. 246, pp. 57-63, 2014.

ro

[22] M. Hu, X. Gao, L. Weng, J. Sun and W. Liu, "The microstructure and improved mechanical

re

Surface Science, vol. 313, pp. 563-568, 2014.

-p

properties of Ag/Cu nanoscaled multilayer films deposited by magnetron sputtering," Applied

lP

[23] W. Bai, J. Cai, X. Wang, D. Wang, C. Gu and J. Tu, "Mechanical and tribological properties of aC/a-C:Ti multilayer films with various bilayer periods," Thin Solid Films, vol. 558, pp. 176-183,

na

2014.

Jo ur

[24] E. Puchi-Cabrera, M. Staia and A. Iost., "Modeling the composite hardness of multilayer coated systems," Thin Solid Films, vol. 578, pp. 53-62, 2015. [25] Y. Qiu, Q. Bai, E. Fu, P. Wang, J. Du, X. Chen, J. Xue, Y. Wang and X. Wang, "A novel approach to extracting hardness of copper/niobium (Cu/Nb) multilayer films by removing the substrate effect," Materials Science and Engineering: A, vol. 724, pp. 60-68, 2018. [26] V. Moorthy, B. Shaw and J. Evans, "Evaluation of tempering induced changes in the hardness profile of case-carburised EN36 steel using magnetic Barkhausen noise analysis," NDT & E International, vol. 36, no. 1, pp. 43-49, 2003. [27] M. Blaow, J. Evans and B. Shaw, "Effect of hardness and composition gradients on Barkhausen

22

Journal Pre-proof emission in case hardened steel," Journal of Magnetism and Magnetic Materials, vol. 303, no. 1, pp. 153-159, 2006. [28] J. Isselin, A. Iost, J. Golek, D. Najjar and M. Bigerelle, "Assessment of the constitutive law by inverse methodology: Small punch test and hardness," Journal of Nuclear Materials, vol. 352, no. 1-3, pp. 97-106, 2006. [29] A. Iost, G. Guillemot, Y. Rudermann and M. Bigerelle, "A comparison of models for predicting the

of

true hardness of thin films," Thin Solid Films, vol. 524, pp. 229-237, 2012.

ro

[30] J. Lesage, D. Chicot, L. Nosei and J. Feugeas, "Prediction of hardness–depth profile from

-p

indentations at surface of materials," Surface Engineering, vol. 25, no. 2, pp. 93-96, 2009.

re

[31] V. Horák, V. Hrubý and T. Mrázková, "Model for Predicting Microhardness Profiles of Steel after

lP

Nitriding," Advances in Military Technology, vol. 8, no. 2, pp. 5-12, 2013. [32] H. Ichimura, Y. Ishii and A. Rodrigo, "Hardness analysis of duplex coating," Surface and Coatings

na

Technology, vol. 169, pp. 735-738, 2003.

Jo ur

[33] Y. Lin, J. Lu, L. Wang, T. Xu and Q. Xue, "Surface nanocrystallization by surface mechanical attrition treatment and its effect on structure and properties of plasma nitrided AISI 321 stainless steel," Acta Materialia, vol. 54, no. 20, pp. 5599-5605, 2006. [34] S. Kossman, A. Iost, D. Chicot, D. Mercier, I. Serrano-Muñoz, F. Roudet, P. Dufrénoy, V. Magnier and A. L. Cristol, "Mechanical characterization by multiscale instrumented indentation of highly heterogeneous materials for braking applications," Journal of Materials Science, vol. 54, no. 6, pp. 4647-4670, 2019. [35] J. Swadener, E. George and G. Pharr, "The correlation of the indentation size effect measured with indenters of various shapes," Journal of the Mechanics and Physics of Solids, vol. 50, no. 4,

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Journal Pre-proof pp. 681-694, 2002. [36] Y. Milman, А. Golubenko and S. Dub, "Indentation size effect in nanohardness," Acta Materialia, vol. 59, no. 20, pp. 7480-7487, 2011. [37] J. Balko, T. Csanádi, R. Sedlák, M. Vojtko, A. KovalĿíková, K. Koval, P. Wyzga and A. NaughtonDuszová, "Nanoindentation and tribology of VC, NbC and ZrC refractory carbides," Journal of the European Ceramic Society, vol. 37, no. 14, pp. 4371-4377, 2017.

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[38] W. Nix and H. Gao, "Indentation size effects in crystalline materials: A law for strain gradient

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plasticity," Journal of the Mechanics and Physics of Solids, vol. 46, no. 3, p. 411–425, 1998.

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[39] Y. Gao, H. Xu, W. Oliver and G. Pharr, "Effective elastic modulus of film-on-substrate systems

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under normal and tangential contact," Journal of the Mechanics and Physics of Solids, vol. 56,

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no. 2, pp. 402-416, 2008.

[40] Y.-H. Lee, W.-j. Ji and D. Kwon, "Stress Measurement of SS400 Steel Beam Using the Continuous

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Indentation Technique," Experimental Mechanics, vol. 44, pp. 55-61, 2004.

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[41] L. Shen, Y. He, D. Liu, Q. Gong, B. Zhang and J. Lei, "A novel method for determining surface residual stress components and their directions in spherical indentation," Journal of Materials Research, vol. 30, no. 8, pp. 1078-1089, 2015. [42] D. Chicot, "Hardness length-scale factor to model nano- and micro-indentation size effects," Materials Science and Engineering A, vol. 499, p. 454–461, 2009. [43] G. Farges and D. Degout, "Interpretation of the indentation size effect in Vickers microhardness measurements-absolute hardness of materials," Thin Solid Films, vol. 181, pp. 365-374, 1989. [44] A. lost and R. Bigot, "Indentation size effect: reality or artefact?," Journal of Materials Science,

24

Journal Pre-proof vol. 31, pp. 3573-3577, 1996. [45] H. Abib, A. Iost, A. Montagne, K. Rahmoun, B. Ayachi and J. Vilcot, "Investigations on the mechanical properties of the elementary thin films composing a CuIn1− xGaxSe2 solar cell using the nanoindentation technique," Thin Solid Films, vol. 633, pp. 71-75, 2017.

Appendix A

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Direct decomposition of modeling function from the integral approach

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A1: Theory

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A parametric form for 𝐻𝑐 (ℎ) is chosen as a differentiable function 𝐻𝑐𝑚𝑜𝑑 (ℎ, 𝜃) (at least to order 2). 𝑒𝑥𝑝

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The objective function is set to be: 𝑤(𝜃) = ∑𝑛𝑖=1(𝐻𝑐𝑚𝑜𝑑 (ℎ𝑖 , 𝜃) − 𝐻𝑐

2

(ℎ𝑖 )) to estimate the optimal

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𝑒𝑥𝑝 parameters 𝜃̂. Thus, a function 𝐻𝑐 (ℎ, 𝜃̂) is defined, and to simplify the notation, it will be simply

denoted as 𝐻𝑐 (ℎ). The objective is to find 𝛱(𝑝). In this case, a direct decomposition is possible. The

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Eq. (4) is written:

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ℎ ℎ 1 2 ∙ ℎ ∙ 𝐻𝑐 (ℎ) = ℎ ∫ 𝛱(𝑝)𝑑𝑝 − ∫ 𝑝 ∙ 𝛱(𝑝)𝑑𝑝 2 0 0

(A1)

A first derivation of the equation above gives: ℎ 1 ℎ ∙ 𝐻𝑐 (ℎ) + ∙ ℎ2 ∙ 𝐻𝑐′ (ℎ) = ∫ 𝛱(𝑝)𝑑𝑝 2 0

(A2)

A second derivation gives: 1 𝛱(ℎ) = 𝐻𝑐 (ℎ) + 2ℎ ∙ 𝐻𝑐′ (ℎ) + ∙ ℎ2 ∙ 𝐻𝑐′′ (ℎ) 2

(A3)

−𝑖 As an example, if this method is applied to model of 𝐻𝑐 (ℎ) such as 𝐻𝑐 (ℎ) = ∑𝑚 𝑖=0 𝑎𝑖 ∙ ℎ , the Eq.

(A3) gives:

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Journal Pre-proof 𝑚

𝑚

𝑚

−𝑖

𝛱(ℎ) = ∑(𝑎𝑖 ∙ ℎ ) − 2ℎ ∑(𝑎𝑖 ∙ 𝑖 ∙ ℎ 𝑖=0

−𝑖−1

𝑖=1

1 ) − ∙ ℎ2 ∑(𝑎𝑖 ∙ 𝑖 ∙ (−𝑖 − 1) ∙ ℎ−𝑖−2 ) 2 𝑖=2

𝑚

𝑚

𝑖=0

𝑖=0

2 − 3𝑖 + 𝑖 2 −𝑖 (𝑖 − 1)(𝑖 − 2) −𝑖 𝛱(ℎ) = ∑ (𝑎𝑖 ∙ ∙ ℎ ) = ∑ (𝑎𝑖 ∙ ∙ℎ ) 2 2 Here,

replacing

the

𝛱(𝑝) = ∑𝑚 𝑖=0 (𝑎𝑖 ∙

name

(𝑖−1)(𝑖−2) 2

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the

variable does

not

matter,

and

it

is

obtained:

∙ 𝑝−𝑖 ). To avoid divergent improper integrals, this equation cannot be

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defined in zero.

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A.2: Solution to the homogeneous differential equation

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In the Eq. (4), 𝐻𝑐 (ℎ) is known and the function 𝛱(𝑝) is searched. So if Eq. (A3) is considered like a

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differential equation, 𝐻𝑐 (ℎ) may be called “particular solution”, 𝐻𝑐 𝑝 (ℎ). If Eq. (A4) is considered as a homogeneous differential equation, it can be used the mathematical results concerning the

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complete solution.

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1 𝐻𝑐 (ℎ) + 2ℎ𝐻𝑐′ (ℎ) + ℎ2 𝐻𝑐′′ (ℎ) = 0 2

(A4)

particular solution.

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If the Eq. (A4) has a solution, this solution gives a complete solution for 𝛱(ℎ) in addition with the

The equation 𝛱(𝑝) = ∑𝑚 𝑖=0 (𝑎𝑖 ∙ sought.

(𝑖−1)(𝑖−2) ∙ 2

𝑝−𝑖 ) shows that 𝑖 = 1 and 𝑖 = 2 gives the solution

1

1

The general solution can be written as 𝐻𝑐 (ℎ) = 𝑘1 ∙ ℎ + 𝑘2 ∙ ℎ2 . If 𝐻𝑐 (ℎ) is a solution of (A3) then 𝐻𝑐 (ℎ) + 𝐻𝑐 𝑝 (ℎ) is a general solution whatever 𝑘1 and 𝑘2 . The limit conditions can allow calculating 𝑘1 and 𝑘2 . A.3: Limit conditions on the solution of Eq. (A3) The boundary conditions are known:

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Journal Pre-proof lim 𝐻𝑐 (ℎ) = lim 𝛱(𝑝) = 𝐻𝑓

ℎ→0

(A5)

𝑝→0

Where 𝐻𝑓 is the hardness close to the outer surface. It is known a relation between the slope of the hardness and the slope of the function 𝛱(ℎ) for ℎ = 0, it can be written as: 1 𝛱′ (ℎ) = 𝐻𝑐′ (ℎ) + 2𝐻𝑐′ (ℎ) + 2ℎ𝐻𝑐′′ (ℎ) + ℎ. 𝐻𝑐′′ (ℎ) + . ℎ2 . 𝐻𝑐′′′ (ℎ) 2

(A6)

For ℎ = 0, it gives : 𝛱′ (0) = 3𝐻𝑐′ (0). So, if 𝐻′(ℎ = 0) = 0, it is obtained 𝛱′(ℎ = 0) = 0.

lim 𝐻𝑐 (ℎ) = lim 𝛱(𝑝) = 𝐻𝑠 𝑝→+∞

(A7)

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ℎ→+∞

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In the same way, if a higher depth a constant hardness (asymptotic behaviour) exists, it can be set:

The solutions in Eq. (A5) prohibit us from using functions in 1/ℎ or 1/ℎ2 to approximate

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A.4: Application to Jönsson and Hogmark’s model

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experimental data without a piecewise-defined function.

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Being 𝛱(𝑝) a well-known function, now, the interest is to search 𝐻𝑐 (ℎ) with Eq. (A3). A piecewise-

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defined function problem is analyzed, and to achieve it, the continuity of the function 𝐻𝑐 (ℎ) must be ensured. For example, in the Jönsson and Hogmark’s problem with a function 𝛱(𝑝) which have a

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value 𝐻𝑓 until 𝐶. 𝑡 and 𝐻𝑠 after. In the first defined piece, 𝐻𝑐 1 (ℎ), the boundary condition (A5) is considered. To avoid an infinite solution, the only set solution is 𝑘1 = 0 and 𝑘2 = 0, thus 𝐻𝑐 𝑝 (ℎ) = 𝐻𝑓 . In the following piece, 𝐻𝑐 2 (ℎ), considering an asymptotic behavior a higher depths, Eq. (A7), 𝐻𝑐 𝑝 (ℎ) = 𝐻𝑠 . In this second piece, the continuity in 𝐶 ∙ 𝑡 must be ensured: for C0 continuity 𝑘

𝑘

1 2 (𝐻𝑐 2 (ℎ = 𝐶 ∙ 𝑡) = 𝐻𝑓 ), i.e. 𝐻𝑠 + 𝐶∙𝑡1 + (𝐶∙𝑡) 2 = 𝐻𝑓 , and for C continuity (𝐻𝑐 2 ′(ℎ = 𝐶 ∙ 𝑡) = 0), i.e.

𝑘

2𝑘

1 2 − (𝐶∙𝑡) 2 − (𝐶∙𝑡)3 = 0 between the first and the second defined piece. It is found 𝑘1 = −2 ∙ 𝐶. 𝑡. (𝐻𝑠 −

𝐻𝑓 )

and

𝐻𝑐 2 (ℎ) = 𝐻𝑠 +

𝑘2 = 𝐶 2 . 𝑡 2 . (𝐻𝑠 − 𝐻𝑓 ). −2∙𝐶∙𝑡∙(𝐻𝑠 −𝐻𝑓 ) ℎ

+

𝐶 2 ∙𝑡 2 ∙(𝐻𝑠 −𝐻𝑓 ) ℎ2

Thus,

the

complete

solution

is:

, this is the Jönsson and Hogmark’s solution, and it is the

unique solution with the C0 and C1 continuity for the second piece.

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Journal Pre-proof A.5: Application to the multilayer model Considering the reasoning explained in last section, being 𝛱(𝑝) a well-known function the goal is to search 𝐻𝑐 (ℎ) with Eq. (A3). A piecewise-defined function will be defined, where the continuity of the function 𝐻𝑐 (ℎ) must be ensured. For example, in a multilayer coating problem with a function 𝛱(𝑝) which have a value 𝐻1 until 𝑝1 , 𝐻2 between 𝑝1 and 𝑝2 , 𝐻𝑖 between 𝑝𝑖−1 and 𝑝𝑖 , and 𝐻𝑠 after 𝑝𝑛 , with 𝑗

𝑝𝑗 = ∑𝑖=1 𝐶𝑖 ∙ 𝑡𝑖 (𝑗 = 1, ⋯ , 𝑛). As evoked above, in the first defined piece, to avoid an infinite

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solution, the only set solution is 𝑘1 = 0 and 𝑘2 = 0, thus 𝐻𝑐 1 (ℎ) = 𝐻1 . In the second piece, the 𝑘

𝑘

continuity in 𝑝1 must be ensured: for C0 continuity (𝐻𝑐 2 (ℎ = 𝑝1 ) = 𝐻1 ), i.e. 𝐻2 + 𝑝1 + 𝑝22 = 𝐻1 , and 2𝑘2 𝑝13

= 0 between the first defined piece and the

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1

1

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𝑘

for C1 continuity (𝐻𝑐 2 ′(ℎ = 𝑝1 ) = 0), i.e. − 𝑝12 −

1

second defined piece. It is found 𝑘1 = −2 ∙ 𝑝1 ∙ (𝐻2 − 𝐻1 ) and 𝑘2 = 𝑝12 ∙ (𝐻2 − 𝐻1 ). Thus, the −2∙𝑝1 ∙(𝐻2 −𝐻1 ) ℎ

+

𝑝12 ∙(𝐻2 −𝐻1 ) , ℎ2

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complete solution is: 𝐻2 +

this is the unique solution with the C0 and C1

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continuities for the second piece. For the third piece, the continuity in 𝑝2 must be ensured: for C0 𝑘1 𝑝2

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continuity (𝐻𝑐 3 (ℎ = 𝑝2 ) = 𝐻𝑐 2 (ℎ = 𝑝2 )), i.e. 𝐻3 +

+ 𝑘

𝑘2 𝑝22

C1 continuity (𝐻𝑐 3 ′(ℎ = 𝑝2 ) = 𝐻𝑐 2 ′(ℎ = 𝑝2 )), i.e. − 𝑝12 −

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2

= 𝐻2 + 2∙𝑘2 𝑝23

=

−2∙𝑝1 ∙(𝐻2 −𝐻1 ) 𝑝2 ∙(𝐻 −𝐻 ) + 1 22 1 , 𝑝2 𝑝2

2∙𝑝1 ∙(𝐻2 −𝐻1 ) 2∙𝑝12 ∙(𝐻2 −𝐻1 ) − 2 𝑝2 𝑝23

and for

between

the first defined piece and the second defined piece. It is found 𝑘1 = −2 ∙ 𝑝1 ∙ (𝐻2 − 𝐻1 ) − 2 ∙ 𝑝2 ∙ (𝐻3 − 𝐻2 ) and 𝑘2 = 𝑝12 ∙ (𝐻2 − 𝐻1 ) + 𝑝22 ∙ (𝐻3 − 𝐻2 ). Thus, the complete solution is 𝐻3 + −2∙𝑝1 ∙(𝐻2 −𝐻1 )−2∙𝑝2 ∙(𝐻3 −𝐻2 ) 𝑝2 ∙(𝐻 −𝐻 )+𝑝2 ∙(𝐻 −𝐻 ) + 1 2 1 ℎ2 2 3 2 , ℎ

continuities 𝐻𝑖 +

for

the

∑𝑖𝑗=2 −2∙𝑝𝑗−1 ∙(𝐻𝑗 −𝐻𝑗−1 ) ℎ

third +

piece.

The

this is the unique solution with the C0 and C1 complete

2 ∑𝑖𝑗=2 𝑝𝑗−1 ∙(𝐻𝑗 −𝐻𝑗−1 )

ℎ2

solution

of

the

𝑖 𝑡ℎ -piece

𝑗

is:

, with 𝑝𝑗 = ∑𝑖=1 𝐶𝑖 ∙ 𝑡𝑖 (𝑗 = 2, ⋯ , 𝑛), it means, the

model advanced in the work of Puchi-Cabrera et al. [24] after a rearrangement of terms.

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Journal Pre-proof Figure captions: Figure 1: (a) Hardness profile determined from cross-sectional indentation tests, and (b) hardness profile determined from normal indentation tests. Figure 2: Representation of the function parameters used for a Vickers indenter. Figure 3: Representation of the contact pressure exerted on a normal section of the indenter at a distance 𝑝 from top surface. Figure 4: Representation of the contact pressure exerted on a normal section of the indenter by a 𝑗 multilayer-coating and substrate system at a distance 𝑝 from top surface, where 𝑝𝑗 = ∑𝑖=1 𝐶𝑖 ∙ 𝑡𝑖

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Figure 5: (a) The gradient section of the function 𝛱(𝑝) is approximated with 6 layers, (b) comparison between the composite hardness, 𝐻𝑐 (ℎ), obtained by the multilayer approach (6 layers) and the continuous linear gradient modeling of 𝛱(𝑝).

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Figure 6: Representation of the contact pressure exerted on a normal section of the indenter by a graded surface and substrate system, with a 4-parameters function (𝐻1 ,𝐻2 , 𝑝1 , 𝑝2 ). Figure 7: Effect of noise with 𝐻2 = 2.00 GPa for one Monte Carlo simulation.

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Figure 8: Histograms obtained from Monte Carlo simulation for the variation of the surface hardness and the geometric parameters 𝑝 in relation with the hardness of the substrate 𝐻2 . (a) 𝐻2 = 2.00 GPa, (b) 𝐻2 = 2.20 GPa, (c) 𝐻2 = 1.80 GPa

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Figure 9: 3D-Scatter plot showing the correlation between 𝐻1 , 𝑝1 and 𝑝2 , and their orthogonal projection onto the coordinate planes.

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Figure 10: (a) Composite hardness as a function of indentation depth, 𝐻𝑐 (ℎ), performed by indentation on the top of the specimen with loads from 0.1 to 2500 N, (b) Hardness as a function of the distance from the top surface, 𝐻(𝐷), obtained by microindentation tests performed on the crosssection of the sample. For the two figures, the bold squares correspond to experimental data performed by Lesage et al. . Figure 11: (a) Normal hardness obtained by indentation tests on the normal surface (bold squares), 𝐻𝑐 (ℎ), modeled by an erf sigmoid function (continuous curve), (b) erf sigmoid function (continuous curve) proposed to model the hardness profile 𝐻(𝐷) of the experimental data (bold squares) .

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FIG. 1

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profile determined from normal indentation tests.

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Figure 1: (a) Hardness profile determined from cross-sectional indentation tests, and (b) hardness

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FIG. 2

𝑦

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𝑥

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𝑥

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𝑧

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Figure 2: Representation of the function parameters used for a Vickers indenter.

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FIG. 3

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distance 𝑝 from top surface.

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Figure 3: Representation of the contact pressure exerted on a normal section of the indenter at a

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FIG. 4

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Figure 4: Representation of the contact pressure exerted on a normal section of the indenter by a 𝑗

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multilayer-coating and substrate system at a distance 𝑝 from top surface, where 𝑝𝑗 = ∑𝑖=1 𝐶𝑖 ∙ 𝑡𝑖

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FIG. 5

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Figure 5: (a) The gradient section of the function 𝛱(𝑝) is approximated with 6 layers, (b) comparison

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between the composite hardness, 𝐻𝑐 (ℎ), obtained by the multilayer approach (6 layers) and the

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continuous linear gradient modeling of 𝛱(𝑝).

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FIG. 6

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Figure 6: Representation of the contact pressure exerted on a normal section of the indenter by a

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graded surface and substrate system, with a 4-parameters function (𝐻1 ,𝐻2 , 𝑝1 , 𝑝2 ).

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FIG. 7

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Figure 7: Effect of noise with 𝐻2 = 2.00 GPa for one Monte Carlo simulation.

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FIG. 8

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(a). Monte Carlo simulation for 𝐻2 = 200 GPa.

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(b) Monte Carlo simulation for 𝐻2 = 220 GPa.

(c) Monte Carlo simulation for 𝐻2 = 180 GPa.

Figure 8: Histograms obtained from Monte Carlo simulation for the variation of the surface hardness and the geometric parameters 𝑝 in relation with the hardness of the substrate 𝐻2 . (a) 𝐻2 = 2.00 GPa, (b) 𝐻2 = 2.20 GPa, (c) 𝐻2 = 1.80 GPa

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FIG. 9

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Figure 9: 3D-Scatter plot showing the correlation between 𝐻1 , 𝑝1 and 𝑝2 ,and their orthogonal

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projection onto the coordinate planes.

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FIG. 10

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Figure 10: (a) Composite hardness as a function of indentation depth, 𝐻𝑐 (ℎ), performed by indentation on the top of the specimen with loads from 0.1 to 2500 N, (b) Hardness as a function of

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the distance from the top surface, 𝐻(𝐷), obtained by microindentation tests performed on the cross-

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section of the sample. For the two figures, the bold squares correspond to experimental data performed by Lesage et al. .

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FIG. 11

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Figure 11: (a) Normal hardness obtained by indentation tests on the normal surface (bold squares),

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𝐻𝑐 (ℎ), modeled by an erf sigmoid function (continuous curve), (b) erf sigmoid function (continuous

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curve) proposed to model the hardness profile 𝐻(𝐷) of the experimental data (bold squares) .

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Journal Pre-proof Declaration of interests

☒ The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

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☐The authors declare the following financial interests/personal relationships which may be considered as potential competing interests:

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Graphical abstract

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    

Highlights Hardness profile of graded materials is deduced from normal hardness measurements. Hardness profile is computed from a definite integral and from a differential equation. Numerical stability of the innovative model is analyzed by a Monte Carlo simulation. The model describes hardness variations on a thermochemical treated material. Control carburizing- or nitriding- depth could be done with the proposed model.

43