Online Manuals RFEM 6 | Time-Dependent Analysis (TDA)

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Dr. rer. nat. Marc Gebhardt

2026-05-28

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Overview

Theoretical Principles

Time-Dependent Analysis (TDA) Add-on

Input

Results

Rheological Model

Introduction

Rheological models describe the relation between the deformation of a body and the action of a load on it. For this, the fundamental properties of elasticity, viscosity, and plasticity are described using idealized mechanical components: spring, damper, and friction elements. Since these idealized behaviors usually occur in combination in reality, it is necessary to couple them—or, to use an analogy from electrical engineering, connect them.

Basic Models

Spring Element

The spring element behaves in an ideally elastic manner, thus following Hook's law, which is why it is also called a Hooke element. As is well known, a linear relation between stress σ and strain ε is applied via a constant modulus of elasticity ‘'E’':

σ = E \cdot ε

σ	Stress
E	Modulus of elasticity
ε	Strain

Hooke's Law

Damper Element

The damper element behaves ideally viscously, like an ideal fluid—also known as a Newtonian fluid—and is therefore referred to as a Newtonian element. It describes the time-delayed, irreversible deformation in response to action:

σ = η \cdot \dot{ε}

σ	Stress
η	Dynamic viscosity
ε˙	Strain rate

Ideal Viscous Behavior

Friction Element

The friction element, also known as the St. Venant element, behaves as an ideal plastic material. This means that below the yield strength, it behaves like an ideal solid and does not undergo any deformation when a load is applied. Once the yield strength is exceeded, the element deforms irreversibly, thus behaving like an ideal fluid with endless viscosity.

ε = \{\begin{cases} σ < σ_{F} : 0 \\ σ \geq σ_{F} : ε (t) \end{cases}

ε	Strain
σ	Stress
σF	Yield strength
t	Time

Ideal Plastic Behavior

Coupled Models

Basic Principles

By coupling the basic models, behavior that is closer to reality can be achieved. If the basic models are connected in series, they all have the same stress, and the strain results from the sum of the individual strains. Connecting them in parallel causes the opposite effect. That is, the strain of all submodels is the same, and the total stress is obtained by summing them.
Since viscoelastic behavior is particularly relevant for time-dependent behavior involving creep and relaxation, the following discussion will only address the coupled models consisting of spring and damper elements that are relevant in this context.

Kelvin Chain (Generalized Kelvin-Voigt Model)

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

Generalized Kelvin-Voigt Chain Model

Creep is the primary behavior of the Kelvin chain and best suited for this application, which is why it will be discussed in more detail below. A Kelvin-Voigt element behaves similarly to a sponge in oil: the spring wants to extend, but the oil resists it. After load relief, the spring pulls the sponge back. The model does not provide an immediate elastic response and asymptotically approaches a limit strain. Recovery upon removal of the load occurs with a time delay but is 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:

σ = E \cdot ε + η \cdot \dot{ε} \Rightarrow ε (t) = \frac{σ_{0}}{E} (1 - e^{- \frac{E}{η} t})

σ	Stress
E	Modulus of elasticity
ε	Strain
η	Dynamic viscosity
ε˙	Strain rate
t	Time
σ0	Initial stress
e	Euler's number

Kelvin-Voigt Element

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

ε (t) = \frac{σ}{E_{0}} + \sum_{k = 1}^{n} ε_{k} (t) + \frac{σ}{η_{∞}} t

ε	Strain
t	Time
σ	Stress
E0	Modulus of elasticity of a free spring (immediate strain)
k	Current element
n	Number of elements
εk	Strain of the current element
η∞	Dynamic viscosity of the free damper (Newtonian fluid)

Generalized Kelvin Chain Model – Total Strain

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

J (t) = \frac{ε (t)}{σ} = \frac{1}{E_{0}} + \sum_{k = 1}^{n} \frac{1}{E_{k}} (1 - e^{- \frac{t}{τ_{k}}}) + \frac{t}{η_{∞}}

ε	Strain
t	Time
σ	Stress
E0	Modulus of elasticity of the free spring (immediate strain)
k	Current element
n	Number of elements
Ek	Modulus of elasticity of the current element
e	Euler's number
τk=ηk/Ek	Delay time of the current element
η∞	Dynamic viscosity of the free damper (Newtonian fluid)

Generalized Kelvin Chain Model – Total Strain

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 element and a damper element connected in series. This is shown schematically in the following image:

Generalized Maxwell Chain Model

The Maxwell chain is best suited for relaxation. This will thus be discussed in more detail below. Maxwell behaves similarly to a liquid droplet with memory: under constant strain, it slowly relieves the stress. However, the model is only of limited suitability for long-term creep, since the strain increases continuously under a permanent load. When a sudden load is applied, the spring reacts immediately with a strain, and the stress asymptotically approaches zero. When the forced 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:

\dot{ε} = \frac{\dot{σ}}{E} + \frac{σ}{η} \Rightarrow σ (t) = σ_{0} \cdot e^{- \frac{E}{η} t}

ε˙	Strain rate
σ˙	Stress rate
E	Modulus of elasticity
σ	Stress
η	Dynamic viscosity
t	Time
σ0	Initial stress
e	Euler's number

Maxwell Element

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

σ (t) = E_{∞} \cdot ε + \sum_{k = 1}^{n} σ_{k} (t)

σ	Stress
t	Time
E∞	Modulus of elasticity of the free spring (equilibrium stiffness)
ε	Strain
k	Current element
n	Number of elements
σk	Stress of the current element

Generalized Maxwell Chain Model – Total Stress

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

G (t) = \frac{σ (t)}{ε} = E_{∞} + \sum_{k = 1}^{n} E_{k} \cdot e^{- \frac{t}{τ_{k}}}

σ	Stress
t	Time
ε	Strain
E∞	Modulus of elasticity of the free spring (equilibrium stiffness)
k	Current element
n	Number of elements
Ek	Modulus of elasticity of the current element
e	Euler's number
τk=ηk/Ek	Delay time of the current element

Generalized Maxwell Chain Model – Total Stress

Extension for Nonlinear Material Behavior

In order to consider nonlinear material behavior, a friction element is additionally introduced in series within the stress path. This element symbolizes the nonlinear material behavior and can perform modeling of effects, such as plastification or softening due to fracture behavior. Because it is connected in series, 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. This results in visco-(elasto-)plastic behavior. The extension of the coupled rheological models presented (left: Kelvin model; right: Maxwell chain model) is shown in the image below.

Generalized Kelvin and Maxwell chain model with nonlinear material behavior

Generalized Kelvin (Left) and Maxwell Chain Models (Right) with Nonlinear Material Behavior

Parent Chapter

Theoretical Principles

From Analytical Creep Curve, Through Rheological Model, to Time-Dependent Strain State