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2024-04-29

VE0127 | Kelvin-Voigt Material Model

Description

Kelvin-Voigt material model consists of the linear spring and viscous damper connected in parallel. In this verification example there is tested the time behaviour of this model during the loading and relaxation in a time interval 24 hours. The constant force F_x is applied for 12 hours and the rest 12 hours is the material model free of load (relaxation). The deformation after 12 and 20 hours is evaluated. Time History Analysis with Linear Implicit Newmark method is used.

System Properties	Spring	Stiffness	k	100.000	kN/m
System Properties	Damper	Viscous Damping	c	1000000.000	kNs/m
Load		Force	F_x	1.000	kN

Kelvin-Voigt Material Model

Analytical Solution

Kelvin-Voigt material model is a parallel connection of the spring and damper. The strain in the spring ε_e and the strain in the damper ε_v are equal (ε_e=ε_v=ε). Total stress of this model is defined as a sum of the stress in the spring and the damper (σ=σ_e+σ_v). Using constitutive equations the differential equation for the strain due to the applied stress σ_x is as follows:

E ε (t) + η \dot{ε} (t) = σ_{x}

E	Modulus of elasticity
t	Time
eta	Dynamic viscosity

Strain differential equation for the Kelvin-Voigt material model

This differential equation can be solved to obtain deformation of the Kelvin-Voigt material model at specific time.

ε (t) = \frac{σ_{x}}{E} + C e^{- \frac{E}{η} t}

C	Integration constant

Generall solution of the strain differential equation

C is the integration constant based on the initial conditions. For the zero strain at the beginning of loading (ε(0)=0) the strain is as follows:

ε (t) = \frac{σ_{x}}{E} (1 - e^{- \frac{E}{η} t})

Strain of the Kelvin-Voigt material model (zero strain initial condition)

This formulla can be rewritten into the form using force, deformation, stiffness and damping.

u_{x} (t) = \frac{F_{x}}{k} (1 - e^{- \frac{k}{c} t})

Deformation of the Kelvin-Voigt material model (zero deformation initial condition)

The deformation after relaxation can be calculated by means of simillar process with different initial condition:

u_{x} (t) = \frac{u_{x 1}}{e^{- \frac{k}{c} t_{1}}} e^{- \frac{k}{c} t}

u_x1	Deformation at the end of loading (time t₁) - initial deformation for the relaxation

Deformation of the Kelvin-Voigt material model (relaxation)

The results are summarized in the following table.

RFEM and RSTAB Settings

Modeled in RFEM 6.05
Time History Analysis with Time Diagram is used
Linear Implicit Newmark method is used

Results

Quantity	Anylytical Solution	RFEM 6	Ratio
u_x(t=12 h) [mm]	9.867	9.867	1.000
u_x(t=20 h) [mm]	0.554	0.554	1.000

The time behaviour of the deformation u_x can be seen in the following figure.

RFEM 6 results - Time behaviour of the deformation ux

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Verification Example 0127 | 1

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