Indentation of Ceramics with Spheres: a Century After Hertz

Journal J. Am. Ceram. Soc., 81 [8] 1977–94 (1998) Indentation of Ceramics with Spheres: A Century after Hertz Brian R. Lawn* Materials Science and Engineering Laboratory, National Institute of Standards and Technology, Gaithersburg, Maryland 20899 In this article we review the nature and mechanics of dam- I. Introduction age induced in ceramics by spherical indenters, from the INCE Hertz first investigated the beautiful cone-shaped frac- classical studies of Hertz over a century ago to the present Stures produced in contacts between glass lenses in the day. Basic descriptions of continuum elastic and elastic– 1880s (see Hertz’s Collected Papers1), indentation mechanics plastic contact stress fields are first given. Two distinct has become extensively used in the analysis and characteriza- modes of damage are then identified: Hertzian cone cracks, tion of fracture and deformation properties of brittle ceram- in relatively hard, homogeneous materials, such as glasses, ics,2,3 as well as of metals and other materials.4 Indentation single crystals, fine-grain ceramics (tensile, ‘‘brittle’’ damage bears profoundly on a wide range of other mechanical mode); and diffuse subsurface damage zones, in relatively properties, such as strength, toughness, and wear. Such damage tough ceramics with heterogeneous microstructures (shear, is now recognized as a key limiting factor in the lifetime of ‘‘quasi-plastic’’ mode). Ceramographic evidence is pre- ceramics in many engineering applications,2,3,5 especially bear- sented for the two damage types in a broad range of ma- ings6 and engine components,7 in both monoliths and coatings. terials, illustrating how an effective brittle–ductile transi- Indentation damage applications extend to areas as diverse as tion can be engineered by coarsening and weakening the dental restorations8 and the anthropological study of ancient grain structure. Continuum analyses for cone fracture and tools.9 Accordingly, it is timely in this centennial feature article quasi plasticity, using Griffith–Irwin fracture mechanics to review the nature and mechanics of contact damage, with and yield theory, respectively, are surveyed. Recent micro- due attention to both its rich historical basis in brittle fracture mechanical models of the quasi-plastic mode are also con- and its topical applications in the latest generation of tough sidered, in terms of grain-localized ‘‘shear faults’’ with ex- ceramics. tensile ‘‘wing cracks.’’ The effect of contact-induced Traditionally, Hertzian cone cracks have been most widely damage on the ensuing strength properties of both brittle studied in flat silicate glass plate, using spheres of hard steel or and quasi-plastic ceramics is examined. Whereas cone tungsten carbide. Extensions to other brittle solids, single crys- cracking causes abrupt losses in strength, the effect of tals (especially those with the diamond structure) and some quasi-plastic damage is more gradual—so that more het- hard, fine-grain polycrystalline ceramics became more preva- erogeneous ceramics are more damage tolerant. On the lent in the period 1950–1970.3 The Hertzian fracture begins as other hand, quasi-plastic ceramics are subject to acceler- a surface ring crack outside the elastic contact and then, at a ated strength losses in extreme cyclic conditions (‘‘contact critical load, propagates downward and flares outward within a fatigue’’), because of coalescence of attendant microcracks, modest tensile field into a stable, truncated cone configuration. with implications concerning wear resistance and ma- Much of the early work centered on the mechanics of cone chinability. Extension of Hertzian contact testing to novel crack initiation, especially the empirically observed linear re- layer structures with hard, brittle outer layers and soft, lation between critical load and sphere size, so-called Auer- tough underlayers, designed to impart high toughness while bach’s law, dating from 1891.10 Auerbach’s law posed a para- preserving wear resistance, is described. dox, in that it apparently violated the notion that cone cracks should initiate when the maximum tensile stress in the indented body exceeds the bulk strength of the material. Griffith–Irwin fracture mechanics analysis was first introduced in the late D. B. Marshall—contributing editor 1960s to account for this paradox,11 with many refinements and reinventions during the ensuing three decades. More recently, sphere-indentation methods have been ex- Manuscript No. 190382. Received February 13, 1998; approved June 22, 1998. tended to heterogeneous ceramics with weak internal inter- *Member, American Ceramic Society. faces, large and elongate grains, and high internal residual centennialfeature 1977 1978 Journal of the American Ceramic Society—Lawn Vol. 81, No. 8 stresses—i.e., tougher ceramics characterized by R-curves. The R-curve can be due to several toughening mechanisms, but is most commonly associated with energy dissipation by internal friction at sliding grains, platelets or whiskers, or other microstructural elements that ‘‘bridge’’ the crack wake.12–14 Such ceramics ordinarily appear completely brittle in traditional strength tests. However, the very same microstructural features that enhance long-crack toughness tend also to degrade short- crack toughness, compromising such important properties as wear resistance.15 Hertzian fracture tends to be suppressed in these materials—instead, a ‘‘quasi-plastic’’ deformation zone develops in the strong shear–compression region below the contact. Macroscopically, this deformation region resembles the plastic zones that occur in metals.4 Microscopically, however, the damage is altogether different, consisting of an array of ‘‘closed’’ mode II cracks with internal sliding friction (‘‘shear faults’’) at the weak planes within the microstructure.16 At high loads, secondary ‘‘extensile’’ microcracks initiate at the ends of the constrained faults. Much precedent for this type of distributed damage exists in the literature on Fig. 1. Hertzian contact of sphere on flat ceramic specimen. Beyond grossly heterogeneous brittle rocks in confined compression elastic limit, contact initiates cone fracture (‘‘brittle mode’’) or sub- fields.17,18 It is implicit that one can control the degree of quasi surface deformation zone (‘‘quasi-plastic mode’’). plasticity relative to the brittle mode by suitably tailoring the ceramic microstructure.16 Experimental simplicity and amenability to materials evalu- contact radius defines the spatial scale of the contact field. The ation are features of general indentation testing. Sometimes mean contact pressure (P/␲a2 (2 ס indentation is the only practical means of obtaining fundamen- p tal information on critical lifetime-limiting damage modes in 0 some ceramics, particularly the quasi-plastic mode. However, defines the intensity of the contact field. The maximum tensile we will not attempt to cover the entire field of indentation stress in the specimen occurs at the contact circle: testing here, omitting in particular parallel developments in the 1 1970s using Vickers and Knoop indenters (‘‘sharp’’ indenters). ␴ = ͑1 − 2␯͒p (3) These parallel developments have been reviewed else- m 2 0 where.2,3,5 We simply point out here one major advantage of The maximum shear stress is located along the contact axis at ‘‘blunt,’’ spherical indenters—they enable one to follow the a depth ≈0.5a below the surface: entire evolution of damage modes, as a progressive transition 19 ≈ from initial elasticity to full plasticity. ␶m 0.48p0 (4) The layout in this article is as follows. We begin with a consideration of the stress fields beneath a spherical indenter, The mean contact pressure in Eq. (2) can be written in other in both elastic and elastic–plastic contact. Then we present useful forms by combining with Eq. (1). One such form ex- micrographic evidence for the two modes of indentation dam- presses p0 in terms of a and r: (3E/4␲k)a/r (5) ס age, single cone cracking (‘‘brittle mode’’) and distributed mi- p crodamage (‘‘quasi-plastic mode’’), in a broad range of glassy 0 and polycrystalline ceramics. Specific attention is focused on Equation (5) prescribes a linear relation between p0, ‘‘inden- the controlling role of microstructure in the competition be- tation stress,’’ and a/r, ‘‘indentation strain,’’ leading to a pro- tween these two modes. Damage models for the two modes, at cedure for obtaining basic stress–strain information.4,22 An- both the macroscopic and microscopic levels, are described. other useful form is given in terms of P and r: The practical issue of strength degradation from damage accu- 2/3 2 1/3 (3E/4k) (P/␲r ) (6) ס mulation, in particular relation to damage tolerance, is then p0 examined. Finally, the scope of Hertzian contact testing is il- Principal normal and shear stresses are calculable from ana- lustrated by examining the most recent work on contact fatigue lytical solutions of the contact boundary conditions (Panel A). and damage in layer structures. It is conventional to define ␴1 Ն␴2 Ն␴3 nearly everywhere within the Hertzian field, so that ␴1 is the most tensile principal .␴ − ␴ ) is the maximum principal shear stress)1 ס stress and ␶ II. Contact Stress Fields 13 2 1 3 Figure 2 shows contours of ␴ , ␴ , and ␶ (␴ is a ‘‘hoop’’ Consider the frictionless contact of a sphere, radius r,at 1 3 13 2 stress). The ␴1 tensile stresses (shaded) in Fig. 2(a) concentrate normal load P, on a flat continuum specimen, Fig. 1. The field in a shallow surface region, with maximum value ␴ at the is initially elastic. Beyond a critical load, either a Hertzian cone m contact circle (Eq. (3)). Included in Fig. 2(a) are ␴3 stress crack (‘‘brittle solid’’) or a subsurface deformation zone trajectories (dashed lines) from the specimen surface, defining (‘‘plastic solid’’) initiates. Here we outline the basic features paths always normal to ␴1 within the plane of the diagram. The of the stress fields associated with elastic and elastic–plastic rapid decrease of ␴ along these trajectories is a characteristic contacts. 1 feature of contact problems. Note that the ␴3 stresses in Fig. (1) Elastic Fields 2(b) are everywhere compressive.

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