Rheological Model

Introduction

Rheological models describe the relationship between the deformation of a body and the load acting on it. For this purpose, the basic properties of elasticity, viscosity, and plasticity are described with idealized mechanical basic elements: spring, damper, and friction element. Since in reality a combination of these idealized behaviors usually occurs, these must be coupled, or connected as an analogy to electrical engineering.

Basic Models

Spring Element

The spring element behaves ideally elastically, thus following Hooke's law, which is why it is also called the Hooke element. As is known, a linear relationship between stress σ and strain ε is assumed here via a constant modulus of elasticity E:

Damper Element

The damper element behaves ideally viscously, like an ideal fluid, also called a Newtonian fluid, and is therefore referred to as the Newton element. It describes the time-delayed irreversible deformation due to an acting stress:

Friction Element

The friction element, also called the St. Venant element, behaves ideally plastically. This means that it behaves like an ideal solid below the yield strength and does not experience any deformation from an applied load. Once the yield strength is exceeded, the element deforms irreversibly, thus behaving like an ideal fluid with infinitely small viscosity.

Coupled Models

Basics

By coupling the basic models, a more realistic behavior can be achieved. If the basic models are connected in series, they all experience the same stress; the strain then results from the sum of the individual strains. Parallel connection produces the opposite effect. That is, the strain of all sub-models is the same, and the total stress results from summation. Since viscoelastic behavior is particularly relevant for the time-dependent behavior under creep and relaxation, only the coupled models relevant in this regard, consisting of spring and damper elements, will be discussed in the following.

Kelvin Chain (Generalized Kelvin-Voigt Model)

In the so-called Kelvin chain, Kelvin-Voigt elements, optionally with a free spring E₀ (instantaneous strain) and a free damper η_∞ (Newtonian flow), are connected in series. A Kelvin-Voigt element consists of a spring and a damper element, which are connected in parallel. This is shown schematically in the following image:

Creep is the primary behavior of the Kelvin chain and it is best suited for this, which is why it will be discussed in more detail below. Kelvin-Voigt behaves comparably to a sponge in oil: the spring wants to expand, the oil slows it down. After unloading, the spring sucks the sponge back. The model provides no immediate elastic response and asymptotically approaches a limit strain. The recovery deformation upon removal of the load occurs with a time delay, but completely.

The constitutive equation of a Kelvin-Voigt element and the time-dependent strain over time (creep) under constant stress are shown in the following formula:

The total strain results from the superposition of the partial strains according to the following equation:

From this, the total creep function can be derived under constant stress and using the constitutive equation:

Maxwell Chain (Generalized Maxwell Model)

In the Maxwell chain, Maxwell elements, optionally with a free spring E_∞ (equilibrium stiffness), are connected in parallel. A Maxwell element consists of a spring and a damper element, which are connected in series. This is shown schematically in the following image:

The Maxwell chain is best suited for relaxation. This will therefore be discussed in more detail below. Maxwell behaves comparably to a fluid droplet with memory: under constant strain, it slowly relieves itself of the stress. For long-term creep, however, the model is only suitable to a limited extent, as the strain increases continuously under permanent load. In the event of a sudden load application, the spring reacts immediately with a strain; the stress asymptotically approaches zero. When the imposed strain is removed, the elastic component springs back immediately, while the strain component of the damper remains.

The constitutive equation of a Maxwell element and the time-dependent stress over time (relaxation) under constant strain are shown in the following formula:

The total stress results from the superposition of the partial stresses according to the following equation:

From this, the relaxation modulus can be derived under constant strain and using the constitutive equation: