It is commonly observed that, besides their instantaneous response (either reversible or irreversible), materials subjected to a mechanical action also demonstrate a delayed behavior that results, for example, in deformation increasing under the action of a constant load. The phenomenon is well known for fluids, with the concept of viscosity (either Newtonian or non-Newtonian), and is also observed for solid materials such as rocks, metals, polymers, etc. Acknowledging the fact that the corresponding timescales are of completely different orders of magnitude depending on the concerned material, this phenomenon may be considered as illustrating the popular aphorism by Heraclitus of Ephesus: “Panta rhei” – “Everything flows”¹.

The theory of viscoelasticity has been built up as a mechanical framework for modeling important aspects of the delayed behavior of a wide range of materials. The presentation proposed here is guided by standard practical applications in various domains of civil or mechanical engineering. It is therefore essentially devoted to linear viscoelastic behavior within the small perturbation framework and does not cover the case of large viscoelastic deformations.

The first part of the book, namely Chapters 1–3, is dedicated to the one-dimensional viscoelastic behavior modeling with the meaning that, for the considered constituent material element or the system under concern itself, both the action it is subjected to and the consequent response can be modeled as one-dimensional.

Within this simple mechanical framework, Chapter 1 introduces the fundamental concepts of viscoelastic behavior from the phenomenological viewpoint of the basic creep and relaxation tests. The viscoelastic constitutive equation can be written as a functional relationship between the action and response histories. It is linear in the case of linear viscoelastic behavior, which is often defined through Boltzmann’s superposition principle, and takes the form of Boltzmann’s integral formulas whose kernels are derived from the creep and relaxation functions. For a non-aging material, these formulas can be identified as Riemann’s convolution products, which call for the use of Laplace or Laplace–Carson transforms, with operational calculus substituting computations in the convolution algebra with ordinary algebraic calculations. It is worth noting that the linearity and non-aging assumptions are here introduced separately and independently from each other, as it has been observed that, in many practical cases (e.g. civil engineering), linearity may be taken as a convenient simplifying assumption valid within a given range of applications, while material aging shall be taken into account.

Rheological models are commonly used as a thought support when trying to write simple one-dimensional constitutive laws matching experimental results. The most classical ones are presented in Chapter 2 in the case of non-aging linear viscoelastic behavior. The typical viscoelastic response to a harmonic loading process is illustrated by the example of the standard linear solid.

Chapter 3 is devoted to the analysis and solution of some illustrative quasi-static evolution problems. It is underscored that pre-eminence and priority must be given to an in-depth physical (and practical) understanding of the problem at hand before entering the mathematical treatment step. Stating the loading process and history properly is essential to reach a correct description and anticipation of the phenomena that will actually take place. It is shown that, for many practical problems, using Boltzmann’s integral operator makes it possible to straightforwardly derive the solution to the problem at hand from its counterpart within the linearized elastic framework. Particular attention is given to the potentially damaging consequences of creep and relaxation phenomena on prestressed structures if they are not correctly anticipated.

This concludes the first part of the book, which may be sufficient for a first analysis of many practical cases, provided that the action and response variables are adequately defined.

The second part of the book, namely Chapters 4–6, discusses the three-dimensional issue within the framework of classical continuum mechanics and the small perturbation hypothesis.

In Chapter 4, relying on the physical concepts introduced in the one-dimensional case, fundamental creep and relaxation experiments are again introduced with the necessity of describing and defining them more precisely. The linear viscoelastic constitutive equation is then written in terms of tensorial integral operators, whose kernels are the tensorial creep and relaxation functions determined through the basic experiments. These functions must comply with material symmetries which specify them, reducing the number of their scalar components. In the isotropic case, it can be observed that two scalar creep functions, or conversely two scalar relaxation functions, are sufficient to completely define the constitutive law in the same way as for linear elasticity. The relationships between these functions bear some similarity with their elastic counterpart, but for the fact that they take the form of integral equations through Boltzmann’s operator.

Quasi-static viscoelastic processes are stated in Chapter 5 within the three-dimensional context and the small perturbation hypothesis. They are defined in the same way as elastic equilibrium problems through field and boundary data depending on the time variable, which must be compatible with the quasi-static equilibrium assumption. It often happens that these data depend on a finite number of scalar loading or kinematic parameters, which may be used to express the global viscoelastic behavior of the studied system. As in the elastic case, no purely deductive method can be proposed for the solution of such problems in a systematic way. Based on intuition and experimental observations, or by analogy with similar linear elastic equilibrium problems, solutions are obtained following the methods that can be qualified as displacement history or stress history methods. Here again, and even more than in the one-dimensional case, an in-depth physical understanding of the problem at hand is necessary for a proper modeling of the process before any mathematical treatment.

In connection with the typical case studies presented in Chapter 3, a few classical three-dimensional quasi-static viscoelastic processes are examined, which concern popular practical problems in the case of a homogeneous isotropic material. Without any non-aging assumption, explicit solutions are obtained, expressed in a simple way using well-chosen creep or relaxation functions of the material. As a result, the global viscoelastic behavior of the system under concern is expressed by creep and relaxation functions straightforwardly derived from the material ones.

It will be observed that, but for a mere allusion in Chapter 2, thermodynamics is not mentioned anywhere in the book. We plead guilty for what may be considered an obvious shortcoming, especially as regards three-dimensional linear viscoelastic modeling, with the argument that this short book aims to make a first reader familiar enough with viscoelastic phenomena as to “feel” them. With the unfortunate experience that thermodynamics, if introduced too early, may act as a deterrent, we believe that the interested reader will refer to the many comprehensive textbooks that are listed in the bibliography. Furthermore, despite the book being only concerned with quasi-static processes, we hope that it may spur the reader towards the analysis of dynamic processes and wave propagation.