11.7: Summary
- Page ID
- 31526
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)A coating on a planar substrate is an important special case of a composite system. Attention is concentrated here on the mechanical (stress-related) effects that commonly arise, with the possibility of stresses (in-plane only) and strains within both constituents being generated in a variety of ways. In the treatment presented here, edge effects are ignored, which simplifies matters, but considerable attention is paid to the way in which the system may become curved, and to the relationship between the curvature and the internal stresses and strains.
Two main cases are considered, depending on whether it is possible for the assumption to be made that the coating is very much thinner than the substrate. If this is the case, then certain assumptions can be made and some relatively simple equations can be used to describe the behaviour. On the other hand, it is emphasized that there are many practical cases for which this condition cannot be assumed, although (slightly more complex) analytical treatments can then be employed and these are described here in some detail.
It is also worth noting that a further distinction can be drawn, depending on whether the stress state within the coating (and substrate) comprises a uniaxial (in-plane) stress or an equal biaxial set of (in-plane) stresses, with the latter being much more common. In this case, it is explained that the ratio of stress to strain in any given (in-plane) direction is given by the biaxial modulus, rather than the Young’s modulus (for a uniaxial stress state).
Finally, a brief outline is presented of how the stored elastic strain energy associated with the presence of stresses in a coating (and substrate) constitutes a driving force for spallation (interfacial crack propagation). A simple criterion is presented for advance of such a crack and some examples are given of how this can be utilized in some practical cases.
Going further
Books
There are not really any books that specifically cover all of the material in this TLP, although there are, of course, books that treat various aspects of coatings, and of surface engineering more generally. The three below respectively present a review of developments since the original Stoney equation, modelling of progressive deposition of a coating onto a thin substrate and the anisotropy that arises with thin films on (cubic) single crystal substrates.
MR Begley & JW Hutchinson, The Mechanics and Reliability of Films, Multi-layers and Coatings, CUP (2017) ISBN: 9781107131866.
J Mencik, Mechanics of Components with Treated or Coated Surfaces, Springer (2010) ISBN-13: 978-9048146116.
L. B. Freund & S. Suresh, Thin Film Materials: Stress, Defect Formation and Surface Evolution, CUP (2004) ISBN-1139449826
Other resources
There are many journal papers that cover the issue of stresses and strains in surface coatings and also the broader topics of how they can provide various kinds of protection, including thermal, environmental and tribological. The two below respectively present a review of developments since the original Stoney equation and an outline of how progressive deposition of a coating onto a thin substrate can be treated.
GCAM Janssen, MM Abdalla, F van Keulen, BR Pujada & B van Venrooy, Celebrating the 100th Anniversary of the Stoney Equation for Film Stress: Developments from Polycrystalline Steel Strips to Single Crystal Silicon Wafers, Thin Solid Films, 517 (2009) p.1858-1867, https://doi.org/10.1016/j.tsf.2008.07.014
YC Tsui & TW Clyne, An Analytical Model for Predicting Residual Stresses in Progressively Deposited Coatings .1. Planar Geometry, Thin Solid Films, 306 (1997) p.23-33, https://doi.org/10.1016/S0040-6090(97)00199-5
KM Knowles, The Biaxial Moduli of Cubic Materials Subjected to an Equi-biaxial Elastic Strain, J. of Elasticity, 124 (2016) p.1-25, https://doi.org/10.1007/s10659-015-9558-x