Characteristics of local compliance of an elastic body under a small punch indented into the plane part of its boundary
ABSTRACT An asymptotic solution of the contact problem of an elastic body indented (without friction) by a circular punch with a flat base is obtained under the assumption of a small relative size of the contact zone. The resulting formulas involve integral characteristics of the elastic body, which depend on its shape, dimensions, fixing conditions, Poisson's ratio, and location of the punch center. These quantities have the mechanical meaning of the coefficients of local compliance of the elastic body. Relations that, generally, reduce the number of independent coefficients in the asymptotic expansion are obtained on the basis of the reciprocal theorem. Some coefficients of local compliance at the center of an elastic hemisphere are calculated numerically. The asymptotic model of an elastic body loaded by a point force is discussed.
- SourceAvailable from: A.U. Daniels[Show abstract] [Hide abstract]
ABSTRACT: In recent years, dynamic indentation tests have been shown to be useful both in identification of mechanical properties of biological tissues (such as articular cartilage) and assessing their viability. We consider frictionless flat-ended and spherical sinusoidally-driven indentation tests utilizing displacement-controlled loading protocol. Articular cartilage tissue is modeled as a viscoelastic material with a time-independent Poisson's ratio. We study the dynamic indentation stiffness with the aim of formulating criteria for evaluation the quality of articular cartilage in order to be able to discriminate its degenerative state. In particular, evaluating the dynamic indentation stiffness at the turning point of the flat-ended indentation test, we introduce the so-called incomplete storage modulus. Considering the time difference between the time moments when the dynamic stiffness vanishes (contact force reaches its maximum) and the dynamic stiffness becomes infinite (indenter displacement reaches its maximum), we introduce the so-called incomplete loss angle. Analogous quantities can be introduced in the spherical sinusoidally-driven indentation test, however, to account for the thickness effect, a special approach is required. We apply an asymptotic modeling approach for analyzing and interpreting the results of the dynamic spherical indentation test in terms of the geometrical parameter of the indenter and viscoelastic characteristics of the material. Some implications to non-destructive indentation diagnostics of cartilage degeneration are discussed.European Journal of Mechanics - A/Solids 01/2012; 37:304-317. · 1.59 Impact Factor
- [Show abstract] [Hide abstract]
ABSTRACT: Widely used the Bulychev–Alekhin–Shorshorov relation for analyzing nanoindentation load–displacement data to determine elastic modulus of a thin specimen does not account for the size of specimen since the BASh relation is based on analytical solutions of the contact problems for an elastic half-space. In order to model the substrate effect, the unilateral contact problem for a spherical indenter pressed against an elastic layer on an elastic half-space is analyzed for different types of boundary conditions imposed at the interface between the specimen and the substrate. Approximate (asymptotically exact) solutions are obtained in explicit form. The influence of the substrate effect on the incremental contact stiffness is described in terms of the asymptotic constants possessing information about the thickness of the specimen and depending on the relative stiffness of the substrate.Mechanics of Materials - MECH MATER. 01/2010; 42(8):807-815.
Journal of Applied Mechanics and Technical Physics, Vol. 43, No. 1, pp. 147–153, 2002
CHARACTERISTICS OF LOCAL COMPLIANCE
OF AN ELASTIC BODY UNDER A SMALL PUNCH INDENTED
INTO THE PLANE PART OF ITS BOUNDARY
I. I. Argatov
An asymptotic solution of the contact problem of an elastic body indented (without friction) by a
circular punch with a flat base is obtained under the assumption of a small relative size of the
contact zone. The resulting formulas involve integral characteristics of the elastic body, which depend
on its shape, dimensions, fixing conditions, Poisson’s ratio, and location of the punch center.These
quantities have the mechanical meaning of the coefficients of local compliance of the elastic body.
Relations that, generally, reduce the number of independent coefficients in the asymptotic expansion
are obtained on the basis of the reciprocal theorem. Some coefficients of local compliance at the center
of an elastic hemisphere are calculated numerically. The asymptotic model of an elastic body loaded
by a point force is discussed.
Introduction. The need for more accurate calculations of contact pressures between machine elements
stimulates the studies of contact problems of the theory of elasticity for bodies different from a half-space. The
existing solutions for an elastic layer [1, 2], elastic spatial wedge , and elastic truncated sphere [4, 5] are based
on the explicit representation of the Green function. Het´ enyi  and Sheveleva  developed a method of specular
reflection for constructing approximately the Green functions for elastic spatial quarter and octant. Tsvetkov and
Chebakov  and Aleksandrov and Pozharskii  developed a homogeneous-solution method for an elastic plate.
Many numerical algorithms for solving contact problems have been proposed (see, e.g., [10–13]).
Under the assumption of a small contact zone, contact problems can be solved by asymptotic methods [1–3,
5]. Argatov  showed that, to construct several first terms of the asymptotic expansion of the contact-pressure
intensity, it suffices to know several coefficients of the asymptotic expansions of regular components of singular
solutions with singularities corresponding to the point force (Green function), point moments, and polymoments.
In the present paper, these coefficients are interpreted as characteristics of local compliance of an elastic body.
1.Formulation of the Contact Problem for a Punch with a Flat Base.
Matrix. We consider an elastic body occupying a three-dimensional domain Ω. At the boundary of the body,
there is a site Σ, which lies in the Ox1x2plane. Let a circular punch whose center coincides with the coordinate
origin be pressed frictionlessly into Σ. We assume that the radius aεof the punch base ω(ε) is small compared to
the characteristic dimension l of the body Ω. For convenience, we set
where ε is a small positive parameter and a∗is a quantity comparable with l and independent of ε. As l, we take
the radius of the largest sphere with a center at the point O that can be enclosed in the region Ω. Let the body be
fixed along the part of the boundary Γuand stress-free on Γσand Σ outside the contact region.
We denote the resultant vector and resultant moments of the system of forces acting on the punch by F3and
M1and M2, respectively. As a result of loading, the punch moves translationally for the distance δ0and rotates.
Makarov State Marine Academy, St. Petersburg 199106. Translated from Prikladnaya Mekhanika i Tekhnich-
eskaya Fizika, Vol. 43, No. 1, pp. 177–185, January–February, 2002. Original article submitted March 13, 2001.
0021-8944/02/4301-0147 $27.00 c ? 2002 Plenum Publishing Corporation
The rotation is determined by the angles β1and β2. The contact-pressure intensity under the punch p satisfies the
following integral equation (see, e.g., [1, § 19]):
Here G3is the vertical component of the Green vector function with a pole at the boundary point (y1,y2,0).
The unknown quantities δ0, β1, and β2are determined from the equations of equilibrium of the punch
Remark 1. The contact pressure under the punch base should be positive. Skewness of the punch may
lead to detachment of the punch edge from the surface of the elastic body. However, it follows from the solution of
the axisymmetric contact problem for an elastic truncated sphere (see [5, § 5.2.1]) that, in the case of translational
indentation of the punch, the condition of full contact holds for reasonably small values of the parameter ε only.
Since the problem is linear, the quantities δ0, β1, and β2should be related to F3, M1, and M2by the linear
The quantities Πkl(ε) characterize the compliance of the elastic body Ω under the action of the punch with a flat
base ω(ε) applied at the point O at the face of the body Σ. For small ε, the displacement of the punch is mainly
due to the local deformation of the body Ω that occurs in the neighborhood of the punch (according to terminology
of [15, § 133], in the local-perturbation region). We call matrix (1.4) the local-compliance matrix. By virtue of the
reciprocal theorem δ??
One of the goals of the present paper is to construct an asymptotic representation of the matrix Π(ε) as
ε → 0.
2. Asymptotic Modeling of Local Contact Interaction between an Elastic Body and a Punch.
We consider the expansion (see, e.g., [1; 16, § 4.14])
G3(y;x) = T3(x1− y1,x2− y2,x3) + g3(y;x),
where g3(y;x) is the projection of the regular component of the Green vector function onto the vertical axis, T3is
the part of the solution of the Boussinesq problem of an elastic half-space loaded by a unit force (see [16, § 5.11]),
1 − ν2T3(x1− y1,x2− y2,0) =
Here E and ν are Young’s modulus and Poisson’s ratio of the material of the body Ω, respectively.
We use the following asymptotic formula (notation coincides with that in [14, 17]):
G3(y1,y2;x1,x2,0)p(y1,y2)dy1dy2= δ0+ β1x2− β2x1.
p(y)dy = F3,
2, the local-compliance matrix Π is symmetric.
?(x1− y1)2+ (x2− y2)2.
(πE/(1 − ν2))g3(y;x1,x2,0) = A0+ B1x1+ B2x2− A(2)
1+ 2C12x1x2+ C22x2
2) + ... .
Here the dots denote the terms of order O(ε3) [in accordance with (1.1)]. It is noteworthy that, generally, the
coefficients that enter the right side of (2.3) (the method of calculating these coefficients is given in [14, 17]) depend
on the location of the point O (punch center).
Example 1. Let Ω be a layer of thickness h perfectly attached to the rigid base x3= h. In this case, the right
side of (2.3) depends only on the squared distance between the points (y1,y2) and (x1,x2). According to , the fol-
lowing representation in the form of an absolutely convergent power series is valid for?(x1− y1)2+ (x2− y2)2< 2h:
1 − ν2g3(y;x1,x2,0) = −1
h2m[(x1− y1)2+ (x2− y2)2]m.
Comparing this expansion with (2.3), we infer that the only nonzero coefficients are
A0= −a0/h,C11= C22= −a1/h3,
The dimensionless coefficients a0and a1as functions of ν are given in [5, Table 1.2] (see also ). For example,
a0= 1.3769, and a1= −0.6276 for ν = 0.3.
Substituting (2.1)–(2.3) into (1.2) and integrating, we obtain the equation
1 − ν2
?(x1− y1)2+ (x2− y2)2= δ0+ β1x2− β2x1−˜F3A0−˜F3(B1x1+ B2x2)
M1, and ˜
1+ 2C12x1x2+ C22x2
moments, respectively), which are equal to the quantities calculated from formulas (1.3) multiplied by (πE)−1(1−ν2)
M2,nare the normalized polymoments of the distributed contact pressures:
M2are the integral characteristics of the contact-pressure intensity (normalized force and
M2,n=1 − ν2
Equation (2.5) is a so-called “coupled” integral equation of the contact problem for a finite elastic body 
and is a third-approximation equation: the coefficient A0is a first-order correction for the geometry of the elastic
body, the fifth and sixth terms are second-order corrections, and the next terms are third-order corrections.
The method of reducing the integral equation (1.2) of the contact problem for an elastic layer to an ap-
proximate equation by polynomial approximation of the regular component of the integral-operator kernel was
proposed in  (see also [1, § 54]). The properties of the solutions of these equations are discussed in [5, § 1.2].
The term “coupled” means that, after a solution (with undetermined coefficients) of the integral equation of the
contact problem for an elastic half-space is constructed, Eq. (2.5) reduces to a system of linear algebraic equations.
Aleksandrov and Shmatkova  applied this method to the problem of a parabolic punch pressed into an elastic
layer. Argatov  obtained an asymptotic solution of the corresponding nonlinear resulting problem.
3. Solution of the Coupled Equation. Using the results of [20–22], we write the solution of Eq. (2.5) in
π(1 − ν2)
+˜F3(C11+ C22)a2− 2
M2,nshould be related to δ0, β1, and β2. Integration of (3.1) yields
where c = 2π−1a is the translational capacity of the circular punch of base radius a (dependence on the parameter
ε is not indicated).
Calculating the moments of the contact-pressure intensity (3.1) [see the second formula in (1.3)], we obtain
where m = 4(3π)−1a3is the rotational capacity of the circular punch.
Finally, inserting (3.1) into (2.6), we find that
With allowance for (3.5), one can express the quantities ˜
determinant of this system is equal to 1+(3π)−1a3(A(2,0)
However, the exact solution is not necessarily required, since formulas (3.1)–(3.6) themselves are approximate.
4. Asymptotic Representation of the Local-Compliance Matrix. We now perform an asymptotic
analysis of relations (3.4)–(3.6) (see also ). Let the quantities δ0, β1, and β2 be fixed (independent of the
parameter ε). Then, from (3.2) follows the expansion˜F3= ε˜F0
formula (3.1) is derived by retaining terms of order ε3compared to unity [see (2.5), where√
in accordance with the accuracy of Eq. (2.5), we confine ourselves to the approximation
Substitution of (4.1) into (3.2) yields
M2,0 and ˜
) and does not vanish for reasonably small ε = a/l.
M2,2 from Eqs. (3.4) and (3.6). The
3+ ... . It should be borne in mind that
2< εa∗]. Thus,
Using similar reasoning to that used in deriving relation (4.1), we replace the last formula by the following one:
Finally, Eq. (3.3) becomes
3(C11+ C22) +
B2˜F3+ (1/m + B(1)
M1+ (1/m − B(2)
Thus, the force F3and the moments M1and M2are related to the displacements of the punch δ0and its angles
of rotation β1and β2by the approximate equations (4.2) and (4.3). Comparing (4.2) and (4.3) with (1.4), we obtain
the asymptotic formulas for the normalized components of the local-compliance matrix˜Πlk= πE(1 − ν2)−1Πkl:
˜Π00? 1/c + A0+ a2(C11+ C22)/3 + a2(A(2,0)
Formulas (4.4) relate the components of the matrix Π to the capacity characteristics of the punch c and m (which
depend only on the geometry of the punch base) and the coefficients in the asymptotic formula (2.3). The latter
coefficients depend on the shape and size of the elastic body Ω, its fixing conditions, location of the point O, and
Since the matrix Π is symmetric, we have
˜Π22? 1/m − B(2)
˜Π11? 1/m + B(1)
The validity of these equalities can be verified by using the Betti formula and coefficients A(1)
in [14, 17] as coefficients in asymptotic formulas of the type (2.3) for certain singular solutions. However, the
reciprocity relations (4.5) and similar relations can easily be obtained directly from (2.3) with allowance for the
following equality implied by the Betty theorem (see, e.g., [23, Chap. 4, § 3.1]):
g3(y1,y2;x1,x2,0) = g3(x1,x2;y1,y2,0).
Substituting expansion (2.3) into (4.6), we obtain (4.5) and the equalities
Thus, in the general case, the reciprocity relations (4.5) and (4.7) reduce the number of different coefficients in the
asymptotic solution constructed above.
Example 2. For a layer (see Example 1), formulas (2.4), (4.2), and (4.3) can be combined to give
Expansion of expression (4.8) into a series in powers of the parameter ε = a/h yields
1 − ν2aδ0
3(1 − ν2)a3βi
Formulas (4.9) coincide with formulas (48.2) and (50.2) in .
5. Calculation of the Coefficients A0and C11for the Center of an Elastic Hemisphere. Let the
elastic body Ω be shaped like a hemisphere of radius l and fixed over the spherical part of the boundary Γu. We
use the Bubnov–Galerkin method to construct an approximate solution of the problem for a unit force applied to
the center of the cut Σ.
Since the problem of determining the vector function g(0;x) [see (2.1)] is axisymmetric, one can use the
general solution of the Lam´ e equations in cylindrical coordinates r and z, expressed in terms of two harmonic
functions Φ1and Φ2in the Weber form (see [24, § 12]):
gr=1 − ν2
(i = 1,2).
(i = 1,2).
+ 2(1 − ν)Φ2
,gz=1 − ν2
− 2(1 − ν)Φ2
Φ = Φ1+ z∂Φ2
As stress functions, we use the homogeneous harmonic polynomials
iρnPn(cosθ)(i = 1,2),ρ =
cosθ = z/
1for odd n.
(Pnis the Legendre polynomial). Substituting expressions (5.1) and (5.2) into the conditions σzz= 0 and τrz= 0
for z = 0, we eliminate one of the coefficients cn
The homogeneous vector polynomial of the nth degree, which satisfies the homogeneous Lam´ e equations in
the half-space z > 0 and the condition that the stresses vanish at the boundary z = 0, has the components
1= 0 for even n or cn
r= −(1 − α)(1/r)Φn+1+ [2 − (1 + α)(n + 1)](z/r)Φn+ (1 + α)n(z2/r)Φn−1,
z= 2Φn− (1 + α)nzΦn−1,α ≡ ν(1 − ν)−1
for even n and
r= (2/r)Φn+1+ [(1 + α)n − 2](z/r)Φn− (1 + α)n(z2/r)Φn−1,
for odd n. The first three vectors˜ V
z= −(1 − α)Φn+ (1 + α)nzΦn−1
ncalculated by formulas (5.3) and (5.4) have the form
1= rer− 2αzez,
2= −2zrer+ (r2+ 2αz2)ez.
Thus, the cylindrical components of the regular part of the Green vector function G(0;x) satisfy the boundary
1 − ν2gr
1 − ν2gz
for ρ = l and 0 ? θ ? π/2. We use the following approximation of the vector function g(0;x):
In this case, the unknown quantities are calculated by the formulas
2(1 − ν)
sinθcosθ − (1 − 2ν)
1 + cosθ
2(1 − ν)cos2θ
vN(r,z) =1 − ν2
A0= c0/l,C11= C22= c2/l3.
To determine the coefficients c0, c1, ..., and cN, we obtain a system of N + 1 linear algebraic equations using the
condition that the discrepancy in the boundary conditions (5.5) due to approximation (5.6) is orthogonal to each
were performed for N = 2–17. It should be noted that the relative error in determining A0is as small as 2% even
for N = 2. To verify the results obtained, calculations by the method of boundary collocation for equidistant nodes
were also performed.
Conclusions. To refine the asymptotic formula (2.3), it is necessary to use additional parameters that
characterize the geometry of the elastic body. All local-compliance coefficients can be obtained by the complete
asymptotic expansion.Practically, it is possible to obtain explicitly only a few first terms of the asymptotic
representation of the contact pressure (see, e.g., [1, 5]).
By virtue of the reciprocity relations (4.5) and (4.7), expansion (2.3) is simplified:
1, ..., and˜ V
Nover the hemisphere. The calculation results are listed in Table 1. The calculations
(πE/(1 − ν2))g3(y;x1,x2,0) = A0+ Bixi+ Biyi+ Cijxixj+ bijxiyj+ Cijyiyj+ ... .
Here b11= −B(2)
that the linear and quadratic forms appearing in this expression are invariant with respect to rotation of the
coordinate axes implies that, in passing to a new coordinate system, the quantities Biand Cij and bij should be
transformed as a vector and tensors, respectively.
If the body Ω and the parts of the boundary Γuand Γσare symmetric, then b12= 0 and b11= b22at the
point O. However, the last coefficients are determined by solving the corresponding nonaxisymmetric problem.
Formulas (1.4) and (4.4) can be considered as an asymptotic model of an elastic body under point loads
(forces and moments). It is well known (see, e.g., [25; 10, Chap. 10, § 1]) that the singular solution G(0,x) is
meaningless in the neighborhood of the point at which the force is applied. In particular, the displacement at this
point is unbounded, whereas formula (1.4) relates the force F3e3to the generalized displacement δ0e3(compare
with Example 1 in [26, § 8.9]).
It is also known (see [27, Chap. 7, § 21]) that the real distribution law of local loads is difficult to determine,
whereas the resultant force vector is usually known with a high degree of accuracy. A priori knowledge of the
distribution law of local loads is necessary (see [28, p. 301]) if, in calculations, they are reduced to point forces by
passing to the limit (see  and [28, Chap. 3, § 6]). At the same time, to construct an asymptotic model of a
point force, it is necessary to solve the coupled integral equation of the contact problem for an elastic body of finite
dimensions and thereby to determine approximately the pressure under the base of a small punch.
The author is grateful to S. A. Nazarov for useful discussions.
1, b12= b21= B(1)
1, and b22= B(1)
2; summation is performed over repeated indices. The condition
1. I. I. Vorovich, V. M. Aleksandrov, and V. A. Babeshko, Nonclassical Mixed Problems of the Theory of Elasticity
[in Russian], Nauka, Moscow (1974).
2. V. M. Aleksandrov and B. L. Romalis, Contact Problems in Mechanical Engineering [in Russian], Mashinos-
troenie, Moscow (1986).
3. I. A. Lubyagin, D. A. Pozharskii, and M. I. Chebakov, “Indentation of a punch shaped like an elliptic paraboloid
into a spatial elastic wedge,” Prikl. Mat. Mekh., 56, No. 2, 286–295 (1992).
4. V. M. Aleksandrov and D. A. Pozharskii, “Axisymmetric contact problem of a truncated sphere,” Prikl. Mat.
Mekh., 61, No. 2, 305–311 (1997).
5. V. M. Aleksandrov and D. A. Pozharskii, Nonclassical Spatial Problems of Mechanics of Contact Interaction of
Elastic Bodies [in Russian], Faktorial, Moscow (1998).
6. M. Het´ enyi, “A general solution for the elastic quarter space,” Trans. ASME, Ser. E, J. Appl. Mech.,
No. 1, 70–76 (1970).
7. G. I. Sheveleva, “Elastic contact displacements of parts of finite dimensions,” Mashinovedenie, No. 4, 92–98
8. A. N. Tsvetkov and M. I. Chebakov, “Contact problem for a finite body of revolution with a free lateral surface,”
Izv. Akad. Nauk SSSR, Mekh. Tverd. Tela, No. 2, 77–82 (1989).
9. V. M. Aleksandrov and D. A. Pozharskii, “Contact problems for a finite cylinder and circular plate,” Izv. Vyssh.
Uchebn. Zaved., Sev.-Kavkaz. Region, Ser. Estestv. Nauki, No. 1, 33–36 (1999).
10. V. L. Rvachev and V. S. Protsenko, Contact Problems of the Theory of Elasticity for Nonclassical Regions [in
Russian], Naukova Dumka, Kiev (1977).
11. A. S. Kravchuk, “Solution of contact problems using the Green function,” Prikl. Mat. Mekh., 46, No. 2, 283–288
12. G. I. Sheveleva, “Solution of contact problems by the method of successive loading under different equilibrium
conditions,” in: Problems of Mechanical Engineering and Reliability of Machines (collected scientific papers)
[in Russian], No. 4 (1990), pp. 68–74.
13. A. M. Rubin, “Algorithm of the trial method in problems of compressed elastic bodies,” in: Problems of
Mechanical Engineering and Reliability of Machines (collected scientific papers) [in Russian], No. 6 (1993),
14. I. I. Argatov, “Asymptotic solution of the contact problem for a three-dimensional body of finite dimensions,”
Prikl. Mat. Mekh., 63, No. 6, 964–970 (1999).
15. A. E. H. Love, A Treatise on the Mathematical Theory of Elasticity, Dover, New York (1944).
16. V. Novacki, Theory of Elasticity [Russian translation], Mir, Moscow (1975). (PWN, Warszawa, 1983.)
17. I. I. Argatov, “Indentation of a punch shaped like an elliptic paraboloid into a plane boundary of an elastic
body,” Prikl. Mat. Mekh., 63, No. 4, 671–679 (1999).
18. V. M. Aleksandrov and A. A. Shmatkova, “Indentation of a parabolic punch into an elastic layer and of two
parabolic punches into an elastic half-space,” Izv. Ross. Akad. Nauk, Mekh. Tverd. Tela, No. 4, 149–155 (1998).
19. V. M. Aleksandrov, “Some contact problems for an elastic layer,” Prikl. Mat. Mekh., 27, No. 4, 758–764 (1963).
20. V. M. Abramov, “Nonsymmetric indentation of a punch of circular cross section into an elastic half-space,”
Dokl. Akad. Nauk SSSR, 23, No. 8, 759–763 (1939).
21. N. A. Rostovtsev, “Complex potentials in the problem of an in-plan circular punch,” Prikl. Mat. Mekh., 20,
No. 1, 77–82 (1957).
22. V. I. Dovnorovich, Spatial Contact Problems of the Theory of Elasticity [in Russian], Izd. Belorus. Univ., Minsk
23. A. I. Lur’e, Theory of Elasticity [in Russian], Nauka, Moscow (1970).
24. M. A. Koltunov, Yu. N. Vasil’ev, and V. A. Chernykh, Elasticity and Strength of Cylindrical Bodies [in Russian],
Vysshaya Shkola, Moscow (1975).
25. G. N. Savin and V. L. Rvachev, “Displacements caused by a point force,” Prikl. Mekh., 10, No. 2, 222–225
26. Yu. N. Rabotnov, Mechanics of a Deformable Solid [in Russian], Nauka, Moscow (1988).
27. V. V. Novozhilov, Theory of Elasticity, Pergamon Press (1961).
28. V. Z. Parton and P. I. Perlin, Methods of Mathematical Theory of Elasticity [in Russian], Nauka, Moscow (1981).
29. E. Sternberg and R. Yubanks, “The concept of point loads and extension of the range of applicability of
the uniqueness theorem in the linear theory of elasticity,” in: Mechanics (collected scientific papers) [Russian
translation], No. 5 (1956), pp. 56–84.