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

Time-Dependent Analysis (TDA)

Theoretical Principles

Input

Results

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 using idealized mechanical basic elements: spring, dashpot, and friction element. Since in reality, a combination of these idealized behaviors usually occurs, 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 established via a constant modulus of elasticity E:

σ = E \cdot ε

σ	Stress
E	Elasticity modulus
ε	strain

Hooke's Law

Dashpot Element

The dashpot 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:

σ = η \cdot \dot{ε}

σ	Stress
η	Dynamic viscosity
ε˙	strain rate

Ideal Viscous Behavior

Friction Element

The friction element, also called the St. Venant element, behaves ideally plastically. This means that below the yield strength it behaves like an ideal solid and undergoes no deformation from an applied load. Upon exceeding the yield strength, the element deforms irreversibly, i.e., it behaves like an ideal fluid with infinitely small 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

Fundamentals

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. A parallel connection produces the opposite effect. This means the strain of all sub-models is the same and the total stress is obtained by summation. Since for time-dependent behavior under creep and relaxation, the viscoelastic behavior is particularly relevant, only the coupled models relevant in this regard – consisting of spring and dashpot 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 an optional free dashpot η_∞ (Newtonian flow), are connected in series. A Kelvin-Voigt element here consists of a spring and a dashpot element that are connected in parallel. This is shown schematically in the following figure:

Creep is the primary behavior of the Kelvin chain and it is best suited for this, which is why it is 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 instantaneous elastic response and asymptotically approaches a limit strain. The recovery deformation 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) at 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 is obtained 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 the free spring (immediate elongation)
k	Current Element
n	Number of Elements
εk	Strain of the current element
η∞	Dynamic viscosity of the free damper (Newtonian flow)

Generalized Kelvin-Chain Model - Total Strain

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

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	Spannung
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	Retardation time of the current element
η∞	Dynamic viscosity of the free damper (Newtonian flow)

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 here consists of a spring and a dashpot element that are connected in series. This is shown schematically in the following figure:

The Maxwell chain is best suited for relaxation. Therefore, this is 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 to a limited extent for long-term creep, as the strain increases continuously under sustained load. Under sudden load application, the spring reacts immediately with a strain; the stress asymptotically approaches zero. Upon removal of the imposed strain, the elastic portion springs back immediately, while the strain portion of the dashpot remains.

The constitutive equation of a Maxwell element and the time-dependent stress over time (relaxation) at 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 is obtained 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 current element

Generalized Maxwell Chain Model - Total Stress

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

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 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	Retardation time of the current element

Generalized Maxwell Chain Model - Total Stress

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 the viscoelastic and plastic strain. Thus, a visco-(elasto-)plastic behavior arises. The extension of the presented coupled rheological models (left: Kelvin and right: Maxwell chain model) is shown in the following figure.

Parent Chapter

Theoretical Principles