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 bodies: spring, damper, and friction elements. Since a combination of these idealized behaviors usually occurs in reality, they 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 well 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 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 body below the yield strength and does not undergo any deformation due to an applied load. Once the yield strength is exceeded, the element deforms irreversibly, thus behaving like an ideal fluid with infinitely low viscosity.

Coupled Models

Basics

By coupling the basic models, a behavior closer to reality 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 partial models is the same, and the total stress is obtained by summation. Since viscoelastic behavior is particularly relevant for time-dependent behavior under creep and relaxation, only the coupled models relevant in this regard, consisting of spring and damper elements, will be discussed below.

Kelvin Chain (Generalized Kelvin-Voigt Model)

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

Generalized Kelvin-Voigt Chain Model

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 brakes it. After unloading, the spring sucks the sponge back. The model provides no instantaneous elastic response and asymptotically approaches a limit strain. The recovery upon removal of the load is time-delayed but complete.

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

Generalized Maxwell Chain Model

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 withdraws from the stress. However, the model is only suitable for long-term creep to a limited extent, as the strain increases continuously under permanent load. Upon sudden load application, the spring reacts immediately with a strain; the stress asymptotically approaches zero. Upon removal of the imposed strain, 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:

Extension for Nonlinear Material Behavior

To account for nonlinear material behavior, a friction element is additionally introduced in series in the stress path. This symbolizes the nonlinear material behavior and can represent effects such as plastification or softening due to fracture behavior. Due to the series connection, it experiences the same stress as the rest of the model, and vice versa, and the strain results from the sum of viscoelastic and plastic strain. Thus, a visco-(elasto-)plastic behavior arises. The extension of the presented coupled rheological models (left: Kelvin chain model and right: Maxwell chain model) is shown in the following figure.

Generalized Kelvin (left) and Maxwell chain model (right) with nonlinear material behavior