深度依赖弹性模量：点力下固体凹痕分析

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本文主要探讨了"固体中梯度弹性的压痕问题：点力作用下的应力与位移演化——Part I"。该研究发表于《固态结构》杂志第34卷第19期，1997年，由Elsevier Science Ltd.出版，刊载在Great Britain的Pergamon Press上。作者A.E. Giannakopoulos和S. Suresh来自美国麻省理工学院材料科学与工程系，他们在文中深入分析了当弹性基底的杨氏模量E随深度z变化时，由正常点力引起的压痕过程中的应力分布和位移响应。研究假设除了深度依赖的特性外，材料是线性弹性和局部各向同性的。文章关注的重点在于两种不同情况下的解析解： 1. 幂律函数：首先，作者考虑了杨氏模量E随深度的简单幂律关系，即E = E0 * (z/z0)^k，其中0 < k < 1是一个无量纲指数。这种情况下，他们推导出了与Poission比（衡量材料横向应变对纵向应变的比值）v保持固定的闭合形式解。这种模型有助于理解随着深度增加，材料硬度的变化如何影响压痕过程的力学行为。 2. 其他特定函数：文章还探讨了E对z的其他预设函数形式，可能包括但不限于指数函数、对数函数或多项式等。通过这些函数，研究者探究了更广泛的材料梯度弹性对压痕现象的影响，旨在揭示不同梯度模式下材料响应的复杂性。研究结果不仅提供了理论分析，还包含了数值计算的结果，两者相结合揭示了在实际工程和材料科学中，梯度弹性材料在受到点力压痕时的行为规律，这对于理解和设计具有深度依赖弹性的材料性能至关重要。这项工作对于理解材料微观结构对整体性能的影响以及开发新型复合材料有着重要的实践意义。

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A. E. Giannakopoulos and S. Suresh

Global equilibrium connects the constant C with the applied vertical load

at the surface

origin as

n/2

2nC (cos $)I!‘+

’ cl+ = -P.

(10)

As an example, typical of many types of clay and sand, consider the case of

E =

E,&.

Then for a Poisson ratio, v = 0.4, the non-zero components of the fields are

-1P

_ _cos 3/2

4nR2

uR = 6_iot3,2 cos h

% = ,,niIi3,2 sin 6, (11)

where 4 = 7112 at the surface and 4 = 0 along the z-axis.

The present section is closed by recalling the particular case of

k =

1 (Gibson soil,

E = E,z)

and v = 0.5 (incompressibility). Gibson and Kalsi (1974) and Callandine and

Greenwood (1978) proved that a simple radial field exists in this case and that it follows

the distribution of the homogeneous case. In the spherical coordinate system

(R, 4, O),

was found that the non-zero components of displacements, strains and stresses, respectively,

are

UR =

C/R’, ERR =

-2CjR3,

Eoo = Eg4 =

C/R3, CJRR =

2CEo cos 4

’

(12)

with C = 3P/(4nEo). The displaced volume gives a remarkable relation between indentation

displacement and contact stress, which are simply related with a Winkler type of spring

constant of magnitude 2Eo/3 (Callandine and Greenwood, 1978).

Although the previous results had been known (Holl, 1940

;

Hruban, 1958), the effect

of the Poisson ratio for value above the critical value, l/(k+2), has not been examined thus

far. This effect will be explored in a later section with full field finite element calculations.

3.1.2.

The Flamant and Boussinesq problems for the power law case.

We develop in this

section general solutions for line and point load problems for the power law case. As shown

earlier, if the Poisson ratio v is not related to the power law exponent

according to

k =

l/v - 2(0 d

k <

l), then a simple radial field cannot be constructed for the axisymmetric

Boussinesq problem (point force). However, since the elastic property distribution does not

contain any characteristic length, it is possible to use potential theory to extract closed form

results. We solve first the Flamant problem, i.e., the solution for a vertical line load (plane

strain). The solution to the point load that corresponds to the axisymmetric Boussinesq

problem can be derived from it by recourse to Aleksandrov’s theorem which connects the

axisymmetric and plane problems (Aleksandrov, 1961). The approach of Booker

et al.

(1985) was to integrate directly the equilibrium equations in spherical coordinates. The

following method utilizes a different approach which involves the equilibrium equations in

cylindrical coordinates and the connection between axisymmetric and plane problems. As

shown later, the present results converge to the solution of Booker

et al.

in specific cases.

In the governing two dimensional compatibility equations, the stresses are expressed

in terms of the Airy stress function (e.g., Ching-Hua, 1961). Expressing the compatibility

equation in polar coordinates

(R, $I)?

and assuming a state of radial stress, and using

t The present approach with cylindrical coordinates, albeit more cumbersome than with the spherical coor-

dinates for the power law case, is much more versatile because of its use for arbitrary variations of elastic moduli

(especially for the cases that contain intrinsic lengths such as the exponential model discussed earlier).

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深度依赖弹性模量：点力下固体凹痕分析

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