Arrangement and method for the synchronous determination of the shear modulus and of the Poisson's number on samples of elastically isotropic and anisotropic materials

Abstract

The invention relates to an arrangement and to a method for the synchronous determination of the shear modulus and of the Poisson's number on samples of elastically isotropic and anisotropic materials. In the arrangement, an indenter is movable in parallel with its longitudinal axis (A) in the direction of the surface of a sample such that a force action is exerted on the material by its tip. The force can be determined by a device for measuring this force and the indenter is additionally deflected in translation along at least one further axis. The longitudinal axis (A) of the indenter is aligned at an angle 90 with respect to the surface of the sample and the indenter carries out an upward movement and a downward movement. In this respect, a device is present for calculating the shear modulus G and the Poisson's number v from the contact stiffness k determined in this manner, from the acting normal force P, from the indentation modulus M.sub.S and from the shear modulus-related parameter N.sub.S of the sample while taking account of the respective angle .

Claims

1. A method for the synchronous determination of the shear modulus (G) and of the Poisson's number (v) of an elastically isotropic or anisotropic material, comprising the steps of moving an indenter in a normal direction parallel to the longitudinal axis of the indenter against a surface of the material at an angle with the surface not equal to 90, wherein the indenter has a tip aligned with the longitudinal axis, through which tip the indenter exerts normal force (P.sub.normal) against the material surface, while moving the indenter tip in a lateral direction upward and downward, sequentially, exerting lateral-up force (P.sub.lateral-up) and lateral-down force (P.sub.lateral-down), respectively, against the material surface, measuring P.sub.normal, determining contact stiffness (k), indentation modulus (M.sub.S), and shear modulus-related parameter (N.sub.S) of the material, and calculating the shear modulus (G) and the Poisson's number (v) of the material based on P.sub.normal, k, M.sub.S, N.sub.S, and , wherein is the acute angle formed between the surface of the material and the indenter longitudinal axis.

2. The method of claim 1, wherein the indenter tip moves upward and downward along an identical path between two reversal points during reciprocating motion.

3. The method of claim 1, wherein the indenter tip moves parallel with the longitudinal axis in an oscillating manner with a constant amplitude exerting P.sub.normal against the material surface.

4. The method of claim 1, wherein the indenter tip moves parallel with the longitudinal axis in an sinusoidal oscillating manner with a constant amplitude exerting P.sub.normal against the material surface.

5. The method of claim 1, wherein is an angle in the range 1-5.

6. An arrangement, for performing the method of claim 1, comprising a) an indenter configured for i) moving in a normal direction parallel to the longitudinal axis of the indenter against a surface of a material at a contact angle with the surface not equal to 90, wherein the indenter has a tip aligned with the longitudinal axis, through which tip the indenter can exert a normal force (P.sub.normal) against the material surface, while ii) moving the indenter tip in a lateral direction upward and downward, sequentially, for exerting lateral-up force (P.sub.lateral-up) and lateral-down force (P.sub.lateral-down), respectively, against the material surface, b) a device, cooperating with the indenter, for measuring the P.sub.normal, and c) a device for calculating the shear modulus (G) and the Poisson's number (v) of the material based on the P.sub.normal, k, M.sub.S, N.sub.S, and .

7. The arrangement of claim 6, wherein the indenter is configured to effect a contact angle such that is an angle in the range 1-5.

8. The arrangement of claim 6, wherein the contact angle is adjustable.

9. The arrangement of claim 6, wherein the indenter is further configured for moving the indenter tip in a lateral direction upward and downward, sequentially, exerting lateral-up force (P.sub.lateral-up) and lateral-down force (P.sub.lateral-down), respectively, against the material surface.

10. The arrangement of claim 6, wherein the indenter is further configured for moving the indenter tip in a lateral direction upward and downward, sequentially, exerting lateral-up force (P.sub.lateral-up) and lateral-down force (P.sub.lateral-down), respectively, against the material surface along an identical path between two points of reversal.

11. The arrangement of claim 6, wherein the indenter is further configured for moving parallel longitudinal axis in an oscillating manner, such that the tip exerts P.sub.normal against the material surface in an oscillation matter, and for recording the contact stiffness during the movement.

12. The arrangement of claim 6, wherein the indenter is further configured for moving parallel longitudinal axis in a sinusoidal oscillation manner, such that the tip exerts P.sub.normal against the material surface in a sinusoidal oscillation manner, and for recording the contact stiffness during the movement.

Description

(1) The invention will be explained in more detail by way of example in the following.

(2) There are shown:

(3) FIG. 1 in schematic form, a tip of an indenter movable on a surface of a sample inclined at an angle ;

(4) FIG. 2 a line scan which has been detected with a hexahedral tip of an indenter along the topography of a surface of a sample with a wedge shape in a downward and upward movement;

(5) FIG. 3 a change of the values of the vertical contact stiffness determined as a function of the indenter position along the same path as explained in FIG. 2;

(6) FIG. 4 a diagram of determined effective reduced shear moduli G.sub.R of a sample of silica glass applied over the topographic gradient tan ;

(7) FIG. 5 the results obtained at an inclined surface II for the determined reduced shear modulus and the Poisson's number at different positions, with the surface II being inclined such that the tip of the indenter carries out an upward movement; and

(8) FIG. 6 examples for the inclination possibilities of the indenter and the sample surfaces.

(9) To demonstrate the operating principle, a wedge indenter was utilized to form a recess in a surface of a sample 2 of silica glass. The sample 2 has two inclined surfaces arranged opposite one another. They are an inclined surface I and a further inclined surface II. The gradient of the inclined surfaces tan can be changed or influenced by a change of the plastic deformation and depth of the formed wedge indentation in the sample 2. The angle is in this respect the angle between the horizontal and the respective inclined surface when the longitudinal axis A of the indenter 1 is aligned vertically. In FIG. 1, the angle is the angle between the horizontal and the inclined surface of the sample 2. The longitudinal axis A of the indenter 1 is aligned vertically in this representation so that the indenter 1, as indicated by the arrow, is moved in this direction onto the surface of the sample to exert a force action there. It is furthermore pointed out that the force action can be exerted as a sinusoidal normal force P.sub.normal in an oscillating manner and lateral forces P.sub.lateral-up and P.sub.lateral-down act additionally in an additional movement.

(10) FIG. 2 shows a line scan which has been detected using a hexahedral tip of an indenter 1 along the topography of the wedge indentation. The tip is in contact with the sample 2 at the inclined surface I of the edge between two adjacent surfaces, of the hexahedral tip of the indenter 1. The flat surface of the hexahedral tip is in contact with the sample 2 at the inclined surface II when the indenter tip is moved upward from left to right. The indenter tip moves downward at the inclined surface I and upward at the inclined surface II. When the indenter tip is moved in the opposite direction, that is from right to left, it moves downward at the inclined surface II and upward at the inclined surface I.

(11) FIG. 3 shows changes in the values of the vertical contact stiffness as a function of the indenter position along the line scan. The contact stiffness is detected during the movement. As a result of the asymmetry of the geometry of the indenter tip and of the differences of the forces in the upward movement and downward movement respectively, the contact stiffness can be determined at differently profiled surfaces on a forward and backward movement or an upward and downward movement.

(12) The difference between the calculated contact stiffnesses in the upward and downward movement is proportional to the displacement of the sample surface in the lateral direction and in the normal direction. The difference of the forces acting on the tip in the lateral direction and in the normal direction during the upward and downward movement also behaves in this manner. If the lateral displacement for the downward movement is small, the lateral contact stiffness can be determined using the following equations:
m*k.sub.lat=|(k.sub.down*k.sub.up)/(k.sub.upk.sub.down)|,(1)
m=f(P.sub.normal,,),(2)
where

(13) k.sub.down and k.sub.up are the contact stiffness in the normal force direction for the downward movement and for the upward movement and k.sub.lat is the lateral contact stiffness. The correction factor m is a function of the normal force P.sub.load, is the coefficient of friction and is the inclination angle of the inclined surfaces with respect to the horizontal or is the complementary angle to the angle between the inclined surface of the sample and the longitudinal axis of the indenter 1.

(14) The normal force P.sub.normal can be an experimentally defined constant (in the range between 1 N to 3 N) and is a constant for the respective material when m is a variable parameter which is dependent on the tip alignment of the indenter 1 and on the inclined surface(s). After the lateral contact stiffness k.sub.lat has been determined, the reduced shear modulus G.sub.R can be determined. The reduced shear modulus which takes account of the tip of the indenter 1 and the sample 2 can be determined with
G.sub.R=k.sub.lat/8a(3)
G.sub.R=m*G.sub.R(4)
where GR is the effective shear modulus without the correction factor m and a is the contact radius which has been determined for a predefined indentation depth of the tip calibration function.

(15) The Poisson's number v can be determined by
v=(M.sub.S4N.sub.S)/(M.sub.S2N.sub.S),(5)
1/E.sub.R=((1/M.sub.S)+((1v.sup.2.sub.tip)/E.sub.tip)),(6)
1/G.sub.R=((1/N.sub.S)+((2v.sub.tip)/G.sub.tip)),(7)
for elastically isotropic materials. Here, E.sub.tip, G.sub.tip and v.sub.tip are the elastic properties of the indenter tip. E.sub.R is the reduced E modulus of the combination of indenter tip and sample in the measurement in the normal force direction, M.sub.S is the indentation modulus (M.sub.S=E.sub.S/(1v.sub.S.sup.2), N.sub.S is the shear modulus-related parameter of the sample (N.sub.S=G.sub.S/(2v.sub.S)).

(16) FIG. 4 shows the reduced shear modulus G.sub.R of silica glass, applied over the topographic gradient tan . The filled squares in the diagram reproduce the results of the movement of the edge of the hexahedral tip of the indenter 1 along the inclined surface I and the non-filled squares reproduce the results on the movement with the surface of the hexahedral indenter tip along the inclined surface II on a movement from left to right.

(17) It can be recognized in this respect that the orientation of the hexahedral indenter tip and the gradient of the inclination angle have a great influence on the reduced shear modulus G.sub.R. The value of the correction factor m can be determined using the equation (4) and can be used in equations (1, 3) to determine the shear modulus G.sub.R. With knowledge of the values for E.sub.R and G.sub.R for the respective sample, the Poisson's number v can be determined using the equations (5 to 7).

(18) The procedure as described above can be used with dense organosilica glass (OSG) thin films. A wedge indentation having the inclined surfaces I and II, as shown in FIG. 2, can in this respect be formed in the surface of the OSG film. The reduced shear modulus G.sub.R and the Poisson's number v can be calculated using the equations (1 to 7). FIG. 5 shows the results obtained at the inclined surface II. In this respect, the value of the shear modulus G.sub.R is at 2.450.21 GPa and the value for the Poisson's number v at 0.310.08. Both values determined in this manner in this respect lie very close to the expected values of G.sub.R=2.86 GPa and v=0.3. The shear modulus G and the Poisson's number v can be determined sufficiently exactly using the invention.

(19) FIG. 6 shows examples for arrangements in accordance with the invention. In the example shown in FIG. 6a, an arrangement is shown in which an inclined surface can be adjusted in its inclination angle . FIG. 6b gives examples with mechanically or chemically formed inclined surfaces which can also be formed as a wedge-shaped recess. In FIG. 6c, an arrangement with an adjustable Inclination angle of an indenter 1 is shown.

(20) The longitudinal indenter axis A is not aligned perpendicular to the surface of the sample 2 during the obliquely inclined movement of the indenter tip, as is the case with the prior art.

(21) The inclination angle of the surface of a sample can lie in the range 1 to 5; the correspondent tangent values can be seen from the diagram shown in FIG. 4.

(22) In a further example, the shear modulus and the Poisson's number were determined at a sample of organosilica glass with M=12 GPa.

(23) In this respect, a topographic determination of the of the lateral contact stiffness of the sample was made. An indenter having a tip at which corners and edges are formed was used. The lateral contact stiffness was determined separately in each case on an upward movement and downward movement which has been carried out on the identical path distance between two reversal points. In this respect, a force of 1 N was exerted on the indenter in the axial direction of its longitudinal axis A.

(24) The apparent lateral contact stiffness k.sub.lat can be determined using the equation
k.sub.lat=(k.sub.down*k.sub.up)/(k.sub.downk.sub.up)(8)

(25) To determine the effective reduced shear modulus, the contact surface A.sub.C of the indenter tip on the sample was first determined with A.sub.C=(k*/2 E*) where E* is the reduced E modulus of the sample of around 13 GPa.

(26) Using the contact surface A.sub.C thus determined, the contact radius of the indenter tip on the sample can be calculated and can be determined for determining the apparent reduced shear modulus G.sub.r, as G.sub.r? k.sub.lat/8a.

(27) The surface gradient can be calculated from this topography information.

(28) While taking account of the asymmetry of the indenter tip, the sides which contact the sample surface on the upward and downward movement have different geometrical designs so that different tip geometries have to be taken into account in the respective movements. This has the result that respective different correction factors m have to be taken into account for the upward and downward movement. They can be determined as reference values at a quartz sample.

(29) The reduced shear modulus can be calculated from the apparent shear modulus G.sub.R with G.sub.R=m*G.sub.R. A mean value can be calculated using the matrix for values at different positions thus obtained, with extreme values having been excluded from the calculation. A reduced shear modulus G.sub.R of 2.04 GPa0.5 GPa was determined at 886 positions for this sample. The Poisson's number v can be calculated using v=(M.sub.S4 N.sub.S)/(M.sub.S2 N.sub.S), wherein M.sub.S=12 GPa and N.sub.S have been calculated with N.sub.S=((1/G.sub.R)3.62.sup.12 Pa.sup.1)).sup.1.

(30) The mean value of the Poisson's number v was able to be determined with 0.460.2.

(31) With knowledge of the Poisson's number v and of the reduced shear modulus G.sub.R, the value of the shear modulus can be calculated with G.sub.S=N.sub.S (2v). For the sample of organosilica glass it amounted to G=3.3 GPa1.3 GPa.

(32) A determination of the shear modulus and of the Poisson's number can also take place in this form for other materials than the organosilica glass.