METHOD FOR CHARACTERIZING THE CRACKING MECHANISM OF A MATERIAL FROM THE FRACTURE SURFACE THEREOF

Abstract

The disclosed method includes, from a topographic map showing, for a set of points {x} located in a midplane of the fracture surface, a height of the fracture surface h relative to the midplane: a step of determining, for each point x on the topographic map, a quantity ω.sub.∈ representative of an average difference in height).sub.IδxI≦∈ between the height h of the fracture surface at point x in question and the height h of the fracture surface at one or more points {x+δx} located inside a circle of radius ∈ centered on point x in question, a step of determining, according to a test distance δr, a spatial correlation function C∈ representing a spatial correlation between points {x} and points {x+δx} such as IδxI=δr, and a step of determining a correlation length ξ from the spatial correlation function C∈.

Claims

1. Method for characterizing, in a solid structure having undergone cracking, a fracture surface of this structure resulting from the cracking, the method comprising, based on a topographical map representing, for a set of points {x} situated in a mean plane of the fracture surface, a height h(x)of the fracture surface with respect to the mean plane: p1 a step (12) of determining, for each point x of the topographical map, a variable ω.sub.∈(x) representative of an average height difference δh(x, δx).sub.δx|≦∈ between the height h(x) of the fracture surface at the point x in question and the height h(x+δx) of the fracture surface at one or more points {x+δx} situated within a circle formed in the mean plane, of radius ∈, and centred on the point x in question, the set of variables ω.sub.∈(x) defining a function over the set of points {x}.

2. Method according to claim 1, in which the relationship defining the variable ω.sub.∈(x) comprises a real constant Ω.sub.∈ determined so that an average of the function ω.sub.∈(x) is substantially zero over the set of points {x} of the topographical map.

3. Method according to claim 1, also comprising: a step (13) of determining, as a function of a test distance δr, a spatial correlation function C.sub.∈(δr) representative of a spatial correlation between the function ω.sub.∈(x) determined for the set of points {x}, and the function ω.sub.∈(x+δx) determined for the set of points {x+δx} such that |δx|=δr.

4. Method according to claim 3, also comprising: a step (14) of determining a length, called correlation length ξ from the spatial correlation function C.sub.∈(δr), the correlation length ξ being equal to the maximum distance δr=|δx| separating pairs of points {x; x+δx} beyond which the spatial correlation function C.sub.∈(δr) becomes substantially equal to Ω.sub.∈.sup.2=ω.sub.∈(x).sub.x.sup.2.

5. Method according to claim 1, in which the topographical map represents a height h(x) of the fracture surface with respect to the mean plane for a set of points {x} situated on a straight line belonging to the mean plane, the variable ω.sub.∈(x) being representative, for each point x of the topographical map, of an average height difference δh(x, δx).sub.|δx|≦∈ between the height h(x) of the fracture surface at the point x in question and the height h(x+δx) of the fracture surface at one or more points {x+δx}situated on either side of the point x in question at a distance less than or equal to the radius ∈.

6. Method according to claim 1, in which the topographical map represents a height h(x) of the fracture surface with respect to the mean plane for a set of points {x} distributed in the mean plane in two non-parallel directions.

7. Method according to claim 6, also comprising: a step of determining, as a function of a test distance δr and a direction of analysis θ in the mean plane, a spatial correlation function C.sub.∈,θ(δr) representative of a spatial correlation between the function ω.sub.∈(x) determined for the set of points {x}, and the function ω.sub.∈(x+δx) determined for the set of points {x+δx} situated at the distance δr in the direction of analysis θ.

8. Method according to claim 7, in which a spatial correlation function C.sub.∈,θ(δr) is determined for different directions of analysis θ, the method also comprising: a step of determining a correlation length ξ.sub.θ for each direction of analysis θ from the corresponding spatial correlation function C.sub.∈,θ(δr), the correlation length ξ.sub.θ being equal to the maximum distance δr=|δx| separating pairs of points {x; x+δx} beyond which the spatial correlation function C.sub.∈,θ(δr) becomes substantially equal to Ω.sub.∈.sup.2=ω.sub.∈(x).sub.x.sup.2.

9. Method according to claim 8, also comprising: a step of determining a direction of propagation of the cracking based on variations in the correlation length ξ.sub.θ with the direction of analysis θ.

10. Method according to claim 4, also comprising: a step of determining a toughness K.sub.c of the structure from the correlation length ξ or at least one correlation length ξ.sub.θ.

11. Method according to claim 4, also comprising: a step of determining a cracking velocity ν from the correlation length ξ or at least one correlation length ξ.sub.θ, and on a function ν=g(ξ) linking the correlation length ξ or ξ.sub.θ to the cracking velocity ν.

12. Method according to claim 4, also comprising: a step of determining, for each point x of the topographical map, a variable ω.sub.ξ(x) representative of an average height difference δh(x, δx).sub.|δx|˜ξ between the height h(x) of the fracture surface at the point x in question and the height h(x+δx) of the fracture surface at one or more points {x+δx} situated substantially on the perimeter of a circle of radius ξ or ξ.sub.θ centred on the point x in question, and a step of determining a variable h.sub.ξ corresponding to an average of the variables ω.sub.ξ(x) over the set of points {x}.

13. Method according to claim 12, also comprising: a step of determining a toughness K.sub.c of the structure from the variable h.sub.ξ.

14. Method according to claim 1, in which the radius ∈ of the circle is substantially comprised between one times a spatial resolution of the topographical map in the mean plane and approximately ten times this spatial resolution.

15. Method according to claim 1, in which the fracture surface is divided into several zones, the topographical map representing a height h(x) of the fracture surface with respect to a mean plane in each of the zones, the steps (12, 13, 14) of the method being carried out individually for each of the zones of the fracture surface, so as to provide local information relating to the fracture surface, and in particular a correlation lengths field ξ(x) or ξ.sub.θ(x), a toughness field K.sub.c(x), a cracking velocity field ν(x), and a local direction of cracking field.

16. Method according to claim 1, in which the variable ω.sub.∈(x) is representative, for each point x of the topographical map, of an average height difference δh(x, δx)diamond .sub.|δx|≦∈ between, on the one hand, the height h(x) of the fracture surface at the point x in question, and, on the other hand, the height h(x+δx) of the fracture surface at one or more points {x+δx} situated on a closed curve surrounding the point x in question.

17. Method according to claim 1, in which the variable ω.sub.∈(x) is representative, for each point x of the topographical map, of an average height difference δh(x, δx).sub.|δx|˜∈ between the height h(x) of the fracture surface at the point x in question, and the height h(x+δx) of the fracture surface at one or more points {x+δx} situated substantially on the perimeter of the circle of radius ∈ centred on the point x in question.

18. Method according to claim 1, in which the variable ω.sub.∈(x) is determined, for each point x of the topographical map, by the relationship: ${\omega }_{\epsilon }\left(x\right)=f\left(\frac{1}{2}.Math.\log \left({〈\delta .Math..Math.h^2\left(x,\delta .Math..Math.x\right)〉}_{.Math.\delta .Math..Math.x.Math.~\epsilon }\right)\right)$ where the function α.fwdarw.f(α) is an affine function, where the function α>log(α) is a logarithmic function, where the quantity δh(x, δx) defines the height difference h(x+δx)−h(x), where the sign ˜ indicates a substantially equal quantity, and where the operator δh(x, δx).sub.|δx|˜∈ returns an average value of the height differences for the set of pairs of points {x; x+δx} of the topographical map such that the distance |δx| is substantially equal to the radius ∈.

19. Method according to claim 1, in which the variable ω.sub.∈(x) is determined, for each point x of the topographical map, by the relationship:
ω.sub.∈(x)=f(sign(δh(x, δx).sub.|δx|˜∈)) where the function α.fwdarw.sign(α) is a sign function taking the value 1 if a is strictly positive, −1 if a is strictly negative, and 0 if a is zero, where the quantity δh(x, δx)=h(x+δx)−h(x), and where the operator δh(x, δx).sub.|δx|˜∈ returns an average value for the height differences for the set of pairs of points {x; x+δx} of the topographical map such that the length |δx| is substantially equal to the radius ∈.

20. Method according to claim 1, comprising, prior to step (12) of determining variables ω.sub.∈(x), a step (11) of acquiring a topographical map representing, for a set of points {x} situated in a mean plane of the fracture surface, a height h(x) of the fracture surface with respect to the mean plane, said topographical map being used for step (12) of determining the variables ω.sub.∈(x).

Description

DESCRIPTION OF THE FIGURES

[0049] Other features and advantages of the invention will become apparent from reading the detailed description of non-limitative embodiments, together with the attached drawings in which:

[0050] FIG. 1 shows an example of steps that can be implemented in the method for characterizing a fracture surface according to the invention;

[0051] FIG. 2 shows, in its upper part, examples of topographical maps giving a height of a fracture surface with respect to a mean plane for different examples of materials and, in its lower part, fields ω.sub.∈(x) having average height differences determined from these topographical maps;

[0052] FIG. 3 shows, in the form of graphs, a spatial correlation function of the fields ω.sub.∈(x) shown in FIG. 2;

[0053] FIG. 4 shows, in the form of graphs, spatial correlation functions determined from different relationships for the fields ω.sub.∈(x),

[0054] FIG. 5 shows a spatial correlation function determined for one and the same fracture surface in two different directions;

[0055] FIG. 6 shows the development of a correlation length as a function of a direction of the spatial correlation.

DESCRIPTION OF EMBODIMENTS

[0056] As these embodiments are in no way limitative, variants of the invention can in particular be considered comprising only a selection of the characteristics described hereinafter, in isolation from the other characteristics described (even if this selection is isolated within a phrase containing other characteristics), if this selection of characteristics is sufficient to confer a technical advantage or to differentiate the invention with respect to the state of the prior art. This selection comprises at least one, preferably functional, characteristic without structural details, or with only a part of the structural details if this part alone is sufficient to confer a technical advantage or to differentiate the invention with respect to the state of the prior art.

[0057] FIG. 1 shows an example of steps that can be implemented in the method for characterizing a fracture surface according to the invention. In a first step 11 a topographical map is established for the fracture surface to be analysed. A topographical map represents, for a set of points {x} distributed regularly in a mean plane of the fracture surface in question, a height h(x) of the fracture surface with respect to this mean plane. Examples of topographical maps are illustrated in the upper part of FIG. 2 for fracture surfaces formed by cracking in parts constituted by different materials. The topographical map on the left illustrates the fracture surface of a part made from aluminium, the one in the middle illustrates the fracture surface of a part made from mortar, and the one on the right illustrates the fracture surface of a ceramic part. The points x or x(x; y) are defined with respect to a frame of reference of the mean plane comprising an axis X, corresponding in FIG. 2 to the horizontal axis, and an axis Y, corresponding to the vertical axis. The height h(x) is defined along an axis Z that is orthogonal to the axes X and Y. It is shown in FIG. 2 by a greyscale level; the lower the height, the darker the dot. In order to show these topographical maps and the fields ω.sub.∈(x) in an instructive manner, the heights h(x) and ω.sub.∈(x) have been normalized. All the heights h(x) are brought within a range of values comprised between 0 and 1, where the value 0 corresponds to the lowest height (situated below the mean plane) and the value 1 corresponds to the maximum height (situated above the mean plane). The same procedure is applied to the field ω.sub.∈(x).

[0058] Each topographical map is established according to a suitable acquisition technique for the material in question. In the case in point, the topographical map in FIG. 2 for the part made from aluminium was established from a stereoscopic pair of images obtained by scanning electron microscopy, the topographical map for the part made from mortar was established by optical profilometry, and the topographical map for the ceramic part was established using mechanical profilometry. The spatial resolutions along the axes X and Y, marked dx and dy, are substantially equal and their values are 3 μm (micrometres), 50 μm and 8 μm, for the aluminium, the mortar and the ceramic, respectively. The accuracy of the measurements along the axis of the heights, i.e. along the axis Z, is equal to approximately 1 μm, 10 μm and 0.2 μm for the aluminium, the mortar and the ceramic, respectively.

[0059] In a second step 12 of the method according to the invention, a variable ω.sub.∈(x) is determined for each point x of the topographical map. This variable ω.sub.∈(x) is determined so as to be representative of an average height difference δh(x, δx).sub.δx|≦∈ between, on the one hand, the height h(x) of the fracture surface at the point x in question and, on the other hand, the height h(x+δx) of the fracture surface at one or more points {x+δx} situated within a circle of radius ∈ centred on the point x in question. The length of the radius ∈ is preferably determined as a function of the spatial resolution of the topographical map, i.e. as a function of the scale of roughness of the fracture surface. The radius ∈ is for example comprised between approximately one times the spatial resolution of the topographical map and approximately ten times this resolution. Preferably, the radius ∈ is equal to the spatial resolution dx and/or dy. In the examples of FIG. 2, the length of the radius ∈ is chosen to be equal to the resolutions dx and dy of the topographical readings. Over the set of points {x}, the variables ω.sub.∈(x) define a function. More specifically, they define a two-dimensional spatial function. This function is also called a field of the height differences, in particular when reference is made to its graphical representation.

[0060] The variable ω.sub.∈(x) can be determined in different ways. According to a first embodiment, the variable ω.sub.∈(x) is determined as being equal to the average height difference between the height h(x) of the fracture surface at the point x in question, and the height h(x+δx) of the fracture surface at one or more points {x+δx} situated substantially on the perimeter of the circle of radius ∈ centred on the point x in question. The variable ω.sub.∈(x) is then defined by the relationship:

ω.sub.∈(x)=δh(x, δx).sub.|δx|˜∈  (1)

where δh(x, δx)=h(x+δx)−h(x) and where the operator δh(x, δx).sub.|δx|=∈ returns an average value of the height differences for the set of pairs of points {x; x+δx} of the topographical map such that |δ|˜∈.

[0061] According to a second variant embodiment, the variable ω.sub.∈(x) is determined, for each point x of the topographical map, by the relationship:

[00006]$\begin{array}{cc}{\omega }_{\epsilon }\left(x\right)=\frac{1}{2}.Math.\log \left({〈\delta .Math..Math.h^2\left(x,\delta .Math..Math.x\right)〉}_{.Math.\delta .Math..Math.x.Math.~\epsilon }\right)& \left(2\right)\end{array}$

where the function α.fwdarw.log(α) is a logarithmic function, for example the decimal logarithm function.

[0062] According to a third embodiment, the variable ω.sub.∈(x) is determined, for each point x of the topographical map, by the relationship:

ω.sub.∈(x)=sign(δh(x, δx).sub.|δx|˜∈) (3)

in which the function α␣sign(α) is a sign function taking the value 1 if a is strictly positive, −1 if a is strictly negative, and 0 if a is zero.

[0063] In each of the variant embodiments, the variable ω.sub.∈(x) can be determined so that the function ω.sub.∈(x) is substantially zero over the set of points {x} of the topographical map. A real constant Ω.sub.∈ is subtracted in the relationships (1), (2) and (3). This constant Ω.sub.∈ is defined from the corresponding variable ω.sub.∈(x), by the relationship:

Ω.sub.∈=ω.sub.∈(x).sub.x

The relationships (1), (2) and (3) then become:

[00007]$\begin{array}{cc}{\omega }_{\epsilon }\left(x\right)={〈\delta .Math..Math.h\left(x,\delta .Math..Math.x\right)〉}_{.Math.\delta .Math..Math.x.Math.~\epsilon }-{\Omega }_{\epsilon }& \left(1^{\prime }\right)\\ {\omega }_{\epsilon }\left(x\right)=\frac{1}{2}.Math.\log \left({〈\delta .Math..Math.h^2\left(x,\delta .Math..Math.x\right)〉}_{.Math.\delta .Math..Math.x.Math.~\epsilon }\right)-{\Omega }_{\epsilon }& \left(2^{\prime }\right)\\ {\omega }_{\epsilon }\left(x\right)=\mathrm{sign}\left({〈\delta .Math..Math.h\left(x,\delta .Math..Math.x\right)〉}_{.Math.\delta .Math..Math.x.Math.~\epsilon }\right)-{\Omega }_{\epsilon }& \left(3^{\prime }\right)\end{array}$

[0064] The lower part of FIG. 2 illustrates the fields ω.sub.∈(x) determined for the topographical maps of the upper part. Thus, from left to right, the fields ω.sub.∈(x) are shown for the fracture surfaces of the aluminium, mortar and ceramic parts. These fields are defined in the same frame of reference as the topographical maps and are determined by the relationship (2′). In the representation in FIG. 2, the fields ω.sub.∈(x) are also normalized by bringing each value for ω.sub.∈(x) within a range of values comprised between 0 and 1, where the value 0 corresponds to the minimum height difference, and the value 1 corresponds to the maximum height difference. As can be seen in FIG. 2, the fields ω.sub.∈(x) give information on the form and the dimension of the microcracks which resulted in the formation of the fracture surface. In particular, the lines the intensity of which is the clearest give the form of the microcracks at the moment of their coalescence with the other microcracks or with the main crack.

[0065] In a third step 13 of the method according to the invention, a spatial correlation function C.sub.∈(δr) is determined. This function is determined as a function of a test distance δr. This test distance δr preferably varies between the spatial resolution of the topographical map, and the largest dimension of the topographical map in the mean plane. It is representative of a spatial correlation between the function ω.sub.∈(x) determined for the set of points {x}, and the function ω.sub.∈(x+δx) determined for the set of points {x+δx} such that |δx|=δr. The spatial correlation function C.sub.∈(δr) is written:

C.sub.∈(δr)=ω.sub.∈(x).Math.ω.sub.∈(x+δx).sub.x,|δx|=δr   (4)

[0066] FIG. 3 shows, in the form of graphs, the spatial correlation function C.sub.∈(δr) for the fracture surfaces of the aluminium, mortar and ceramic parts. The x-axis represents the test distance δr, and the y-axis represents the spatial correlation value for this distance. For each material, the correlation function C.sub.∈(δr) was marked for different lengths of radius ∈. In particular, for the aluminium part, the correlation function C.sub.∈(δr) was determined for radii of 3, 6, 9, 15, and 24 μm. The graphs demonstrate the fact that the correlation function C.sub.∈(δr) has a value of zero beyond a certain distance. This distance is called correlation length ξ. It provides information on the length of the damage zone of the cracking process. It should be noted that the correlation function C.sub.∈(δr) has a value of zero beyond the correlation length ξ regardless of the length of the radius ∈, while this length is less than the correlation length ξ. The graphs in FIG. 3 also show that the length of the radius ∈ has a relatively limited influence on the form of the correlation function C.sub.∈(δr). This is particularly true for the distances δr greater than a threshold distance. Thus, the choice of the length of the radius ∈ used for determining the function ω.sub.∈(x) has no critical influence on the remainder of the method, and in particular on the accuracy of the correlation length ξ.

[0067] FIG. 4 shows different spatial correlation functions C.sub.∈(δr) for the fracture surfaces of the aluminium, mortar and ceramic parts. The x-axis represents the test distance δr, and the y-axis represents the spatial correlation value for this distance. For each material, the correlation function C.sub.∈(δr) was determined and plotted for different functions ω.sub.∈(x), denoted (a), (b), and (c), and defined as follows:

[00008]$\begin{array}{cc}{\omega }_{\epsilon }\left(x\right)=\frac{{〈\delta .Math..Math.h\left(x,\delta .Math..Math.x\right)〉}_{.Math.\delta .Math..Math.x.Math.=\epsilon }-{\Omega }_{\epsilon }}{{\sigma }_{\epsilon }}& \left(a\right)\\ {\omega }_{\epsilon }\left(x\right)=\frac{1}{2}.Math.\log \left({〈\delta .Math..Math.h^2\left(x,\delta .Math..Math.x\right)〉}_{.Math.\delta .Math..Math.x.Math.=\epsilon }\right)-{\Omega }_{\epsilon }& \left(b\right)\\ {\omega }_{\epsilon }\left(x\right)={〈\mathrm{sign}\left(\delta .Math..Math.h\left(x,\delta .Math..Math.x\right)\right)〉}_{.Math.\delta .Math..Math.x.Math.=\epsilon }-{\Omega }_{\epsilon }& \left(c\right)\end{array}$

where the quantity σ.sub.∈ denotes the standard deviation of the function ω.sub.∈(x). In the function (a), division by σ.sub.∈ allows the correlation function to be shown on one and the same graph as the correlation functions obtained with (b) and (c). It should be noted that the function (c) could have an alternative form, as follows:

ω.sub.∈(x)=sign(δh(x, δx).sub.|δx|=∈)−Ω.sub.∈

[0068] The method according to the invention comprises a fourth step 14, in which the correlation length ξ is determined from the correlation function C.sub.∈(δr). This step can be carried out in different ways. It consists for example of determining the smallest value of δr for which the value of the correlation function C.sub.∈(δr) is less than a predetermined threshold value. In another embodiment, this step is carried out by selecting a set of points the value of which is greater than a predetermined threshold, substantially equal to ω.sub.∈.sup.2=ω.sub.∈(x).sub.x.sup.2, by plotting a straight line passing as close as possible to the selected points, and by determining the distance δr for which this straight line intersects the straight line of equation C.sub.∈(δr)=Ω.sub.∈.sup.2. FIG. 4 demonstrates the fact that the correlation length ξ is relatively independent of the relationship chosen for the function ω.sub.∈(x).

[0069] FIG. 5 represents a spatial correlation function determined in two different directions for a fracture surface on a part made from mortar. A first direction, denoted X, corresponds to the direction of propagation of the crack. A second direction, denoted Y, corresponds to a direction perpendicular to the direction of propagation, in the mean plane. As can be seen clearly in FIG. 5, the correlation length ξ.sub.X in direction X of propagation of the crack is greater than the correlation length ξ.sub.Y in the direction Y.

[0070] FIG. 6 shows more specifically the variation in the correlation length ξ.sub.θ as a function of the spatial correlation direction. It can be noted that the correlation length ξ.sub.θ passes through a maximum for an angle of zero degrees, corresponding to the direction of propagation of the crack.

[0071] The method according to the invention can comprise additional steps of determining mechanical properties of the material or materials of the structure from the function ω.sub.∈(x), the correlation function C.sub.∈(δr) (or at least one function C.sub.∈,θ(δr)) and/or the correlation length ξ (or at least one correlation length ξ.sub.θ).

[0072] The toughness K.sub.c of a material can in particular be determined from the correlation length ξ or at least one correlation length ξ.sub.θ. By way of example, for a material the damage zone of enlargement L.sub.c of which is described by a cohesive zone characterized by a breaking stress σ.sub.c, the toughness K.sub.c follows the relationship:

[00009]$K_c=\sqrt{\frac{8}{\pi }}.Math.{\sigma }_c.Math.\sqrt{L_c}$

This relationship originates from the publication G. I. Barrenblatt, “The mathematical theory of equilibrium of cracks in brittle solids”, Adv. Appl. Mech. 7, 55 (1962). The correlation length ξ determined from fracture surfaces according to the present invention gives the enlargement L.sub.c of the damage zone, and therefore the toughness of the material, through the relationship:

[00010]$K_c=\sqrt{\frac{8}{\pi }}.Math.{\sigma }_c.Math.\sqrt{\xi }$

The breaking stress σ.sub.c of the material studied can then be determined independently, either by using conventional experimental methods, or by using the values provided in the literature. The breaking stress of a material is generally within the range E/10<σ.sub.c<E/5 where E is the Young's modulus of the material.

[0073] The method can also comprise a step of determining the fracture energy G.sub.c of the material. In fact, the latter follows the relationship G.sub.c=K.sup.2/E which makes it possible to link the fracture energy of the material with the correlation length extracted from the fracture surfaces via the relationship:

[00011]$G_c=\frac{8}{\pi }.Math.\frac{{\sigma }_c^2}{E}.Math.\xi$

[0074] Of course, the invention is not limited to the examples which have just been described and numerous adjustments can be made to these examples without exceeding the scope of the invention. In particular, the various characteristics, forms, variants and embodiments of the invention can be combined together in various combinations if they are not incompatible or mutually exclusive.