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Material degradation due to moisture and temperature. Part 1: mathematical model, analysis, and analytical solutions

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Abstract

The mechanical response, serviceability, and load-bearing capacity of materials and structural components can be adversely affected due to external stimuli, which include exposure to a corrosive chemical species, high temperatures, temperature fluctuations (i.e., freezing–thawing), cyclic mechanical loading, just to name a few. It is, therefore, of paramount importance in several branches of engineering—ranging from aerospace engineering, civil engineering to biomedical engineering—to have a fundamental understanding of degradation of materials, as the materials in these applications are often subjected to adverse environments. As a result of recent advancements in material science, new materials such as fiber-reinforced polymers and multi-functional materials that exhibit high ductility have been developed and widely used, for example, as infrastructural materials or in medical devices (e.g., stents). The traditional small-strain approaches of modeling these materials will not be adequate. In this paper, we study degradation of materials due to an exposure to chemical species and temperature under large strain and large deformations. In the first part of our research work, we present a consistent mathematical model with firm thermodynamic underpinning. We then obtain semi-analytical solutions of several canonical problems to illustrate the nature of the quasi-static and unsteady behaviors of degrading hyperelastic solids.

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Abbreviations

\(\rho \) :

Density of solid in deformed configuration \((\mathrm {kg}\,\mathrm {m}^{-3})\)

A :

Specific Helmholtz potential \((\mathrm {J}\,\mathrm {kg}^{-1})\)

\(\zeta \) :

Dissipation functional \((\mathrm {J}\,\mathrm {kg}^{-1}\,\mathrm {s}^{-1})\)

\(\psi \) :

Strain energy density functional \((\mathrm {J}\,\mathrm {m}^{-3})\)

\(\lambda \), \(\mu \) :

Lamé parameters \((\mathrm {Pa})\)

\(\kappa \) :

Bulk modulus \((\mathrm {Pa})\)

\(\mathbf {u}\) :

Displacement \((\mathrm {m})\)

\(\mathbf {v}\) :

Velocity \((\mathrm {m}\,\mathrm {s}^{-1})\)

\(\vartheta \) :

Temperature \((\mathrm {K})\)

c :

Concentration \((\hbox {l})\)

\(R_s\) :

Specific vapor constant \((\mathrm {J}\,\mathrm {kg}^{-1}\,\mathrm {K}^{-1})\)

\(c_p\) :

Heat capacity \((\mathrm {J}\,\mathrm {kg}^{-1}\,\mathrm {K}^{-1})\)

\(\mathbf {M}_{\vartheta \mathbf {E}}\) :

Thermal expansion tensor \((\mathrm {J}\,\mathrm {m}^{-3}\,\mathrm {K}^{-1})\)

\(\mathbf {M}_{c \mathbf {E}}\) :

Chemical expansion tensor \((\mathrm {J}\,\mathrm {m}^{-3})\)

\(d_{\vartheta c}\) :

Thermo–chemo coupled parameter \((\mathrm {J}\,\mathrm {kg}^{-1}\,\mathrm {K}^{-1})\)

\(\varkappa \) :

Specific chemical potential \((\mathrm {J}\,\mathrm {kg}^{-1})\)

\(\eta \) :

Specific entropy \((\mathrm {J}\,\mathrm {kg}^{-1}\,\mathrm {K}^{-1})\)

\(\mathbf {D}_{\vartheta \vartheta }\) :

Thermal diffusion tensor \((\mathrm {m}^2\,\mathrm {s}^{-1})\)

\(\mathbf {D}_{\varkappa \varkappa }\) :

Diffusivity tensor \((\mathrm {m}^2\,\mathrm {s}^{-1})\)

\(\mathbf {D}_{\vartheta \varkappa }\), \(\mathbf {D}_ {\varkappa \vartheta }\) :

Dufour–Soret effect tensors \((\mathrm {m}^2\,\mathrm {s}^{-1})\)

\(\mathbf {T}\) :

Cauchy stress \((\mathrm {Pa})\)

\(\mathbf {h}\) :

Diffusive flux vector \((\mathrm {kg}\,\mathrm {m}^{-2}\,\mathrm {s}^{-1})\)

\(\mathbf {q}\) :

Heat flux vector \((\mathrm {J}\,\mathrm {m}^{-2}\,\mathrm {s}^{-1})\)

h :

Volumetric source \((\mathrm {kg}\,\mathrm {m}^{-3}\,\mathrm {s}^{-1})\)

q :

Volumetric heat source \((\mathrm {J}\,\mathrm {m}^{-3}\,\mathrm {s}^{-1})\)

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Authors and Affiliations

  1. Department of Civil and Environmental Engineering, University of Houston, Houston, TX, 77204–4003, USA

    C. Xu, M. K. Mudunuru & K. B. Nakshatrala

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  1. C. Xu
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  2. M. K. Mudunuru
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  3. K. B. Nakshatrala
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Correspondence to K. B. Nakshatrala.

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Communicated by Andreas Öchsner.

C. Xu and M.K. Mudunuru are graduate students at University of Houston.

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Xu, C., Mudunuru, M.K. & Nakshatrala, K.B. Material degradation due to moisture and temperature. Part 1: mathematical model, analysis, and analytical solutions. Continuum Mech. Thermodyn. 28, 1847–1885 (2016). https://doi.org/10.1007/s00161-016-0511-4

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