Elastic constants determination of anisotropic materials by depth-sensing indentation

Review Paper Elastic constants determination of anisotropic materials by depth‑sensing indentation Caterina Lamuta1 © Springer Nature Switzerland AG 2019 Abstract Depth-sensing indentation is a useful and powerful tool for the mechanical characterization of materials at the micro and nano scale. This technique allows the determination of the Young modulus from the analysis of the load-penetration depth curve according to specific theoretical models. One of the most used models is that one proposed by Oliver and Pharr. However, when a material with anisotropic mechanical properties is tested, Oliver and Pharr’s theory is no longer suitable to describe the contact mechanics between the indenter tip and the tested material. This paper provides an overview of the theoretical models developed for the evaluation of the elastic constants of anisotropic materials through depth-sensing indentation. Specifically, the cases of generally anisotropic and orthotropic materials are described in order to cover the entire range of anisotropy. Examples on how these models can be applied for the mechanical characterization of generally anisotropic topological insulators and transversely isotropic pyrolytic carbon are also reported. This topical overview represents a useful tutorial for the evaluation of the elastic constants of anisotropic materials by depth-sensing indentation by shading light on the contact mechanics at the micro and nano scale. Keywords Depth-sensing indentation · Indentation modulus · Elastic mechanical properties · Anisotropic materials · Density functional theory · Topological insulators · Pyrolytic carbon 1 Introduction Indentation experiments have been extensively used since 1822 to measure the hardness of materials [1]. The introduction of depth-sensing indentation allowed to measure also the elastic properties of solids (i.e., Young’s modulus) through the real-time monitoring of the applied indentation load and the penetration depth of the indenter tip [2]. Very small volumes of materials, in the sub-micron range, can be tested with depth-sensing indentation, which is one of the most powerful experimental tools for the mechanical characterization of thin films, coatings, nanocomposites, and heterogeneous structures. In a typical depth-sensing indentation tests, an indenter tip is driven into the tested material until a target load or penetration depth is reached. This process is usually performed at constant loading or constant displacement rate. After a short hold period at maximum indentation load, the tip is gradually retracted from the material surface. Tabor [3] and Stilwell and Tabor [4] observed that the elastic modulus of the tested material is related to the displacement recovered during the unloading process and can be calculated from the theory of elasticity. Doerner and Nix [5] modelled the unloading process as a contact problem of a rigid punch on an elastically half space. Using these assumptions, Pharr et al. [6] demonstrated that the indentation modulus M of a tested material is related to the slope of the unloading curve S (i.e., the contact stiffness) through the following relationships: S= dP 2 √ =√ M A dh 𝜋 (1) * Caterina Lamuta, caterina‑ | 1Department of Mechanical Engineering, University of Iowa, Iowa City, IA 52242, USA. SN Applied Sciences (2019) 1:1263 | https://doi.org/10.1007/s42452-019-1301-y Received: 30 May 2019 / Accepted: 17 September 2019 / Published online: 23 September 2019 Vol.:(0123456789) Review Paper SN Applied Sciences (2019) 1:1263 | https://doi.org/10.1007/s42452-019-1301-y where P is the indentation load, h the penetration depth (i.e., the rigid-body displacement of the indenter relative to the half-space), and A the projected contact area. The indentation modulus M, also called reduced Young modulus Er, depends on the elastic properties of the tested material, according to the following equation [6]: anisotropic and orthotropic materials respectively, will be described in this paper. An implementation of these models for the mechanical characterization of generally anisotropic topological insulators and transversely isotropic pyrolytic carbon will be finally described. 2 1 1 1 − 𝜈 2 1 − 𝜈i = = + M Er E Ei (2) where E and ν are the Young modulus and the Poisson ratio of the tested material while Ei and νi are the Young modulus and the Poisson ratio of the indenter material (usually Ei = 1141 GPa and νi = 0.07 for a diamond tip). After S is experimentally measured from the unloading curve, the Young modulus E of a tested material, can then be calculated using Eqs. (1, 2), if the value of the projected contact area A is known. The projected contact area can be obtained from the indenter penetration depth through simple geometrical relationships, such as Eqs. (3, 4), which refer to indenter tips with a spherical and conical geometry, respectively: A = 2𝜋Rhc (3) A = 𝜋h2c tan2 (𝛼). (4) R is the radius of the spherical indenter, α the cone halfangle, and hc is the effective penetration depth. However, during an indentation test, only the maximum penetration depth hmax can be experimentally measured by means of displacement sensors and such value differs from the real penetration depth hc, due to the deflection of the surface of tested sample (a downward deflection is known as “sink-in” while un upward deformation is known as pileup phenomenon) Oliver and Pharr [7] proposed the following equation for the calculation of the real penetration depth, hc: P hc = hmax − 𝜀 max S (5) where ε is the intercept factor, equal to ¾. The measurement of the elastic stiffness through Eqs. (1–5) refers to isotropic materials, characterized by the same mechanical properties in all directions. When anisotropic materials (i.e., materials with different properties along different directions) are tested by depth-sensing indentation, Eq. (2) is no longer suitable to describe the relationship between the indentation modulus M and the elastic constants of such materials. Different theoretical models have been developed to obtain the equivalent of Eq. (2) for anisotropic materials. After a brief introduction on the stress/strain relationship of anisotropic materials, two of the most used models, referred to generally Vol:.(1234567890) 2 Hooke’s law for anisotropic materials The stress/strain relationship (i.e., the Hooke’s law) of generally anisotropic materials with linear elastic behavior can be written as follows: (6) where σij are the stress components, εij the strain components, and Cijkl the elastic moduli. Considering the following symmetry properties [8, 9]: 𝜎ij = Cijkl 𝜀kl 𝜎ij = 𝜎ji (7) 𝜀kl = 𝜀lk (8) (9) the generalized Hooke’s law can be written in matrix form as: Cijkl = Cjikl , Cijkl = Cijlk , Cijkl = Cklij ⎛ 𝜎11 ⎞ ⎡ C11 C12 C13 C14 C15 C16 ⎤⎛ 𝜀11 ⎞ ⎜ 𝜎22 ⎟ ⎢ C12 C22 C23 C24 C25 C26 ⎥⎜ 𝜀22 ⎟ ⎥⎜ ⎜ ⎟ ⎢ ⎟ ⎜ 𝜎33 ⎟ = ⎢ C13 C23 C33 C34 C35 C36 ⎥⎜ 𝜀33 ⎟. ⎜ 𝜎23 ⎟ ⎢ C14 C24 C34 C44 C45 C46 ⎥⎜ 2𝜀23 ⎟ ⎜ 𝜎 ⎟ ⎢ C (...truncated)