1. Date: 22/08/2011
Report Number: 2
Title: self‐healing structures
Keywords: Viscoelastic mechanical models, Maxwell bar, Kelvin bar
Elements in Continuum Mechanics
For simulation the part of elasticity in a material we use a linear spring with constant of Yang's
modulus.
Damper is an element in modeling of viscosity of a material that works with rate of strain.
The last element is friction element that implies in modeling of plasticity of a material, and if stress
amount increases more than yield stress it will affect.
Viscolelasticity
In [1] Maxwell and Kelvin models are described for viscoelastic material (Figure 1), but when I
derived the equations myself I found a hidden assume that it can limited Maxwell model only for
small interval of times and small changes, in continue I begin with assumptions which are used for
equations.
Figure 1. Maxwell model (a) and Kelvin model (b) for modeling viscoelastic materials.
In physical equations for linear springs we can write Hook Law (eq.1) as below:
ܨ ൌ ܭΔܮ (1)

2. And we can extend constant of spring for parallel and cascades configuration respectively (Figure2)
by below forms (eq.2, 3):
Figure 2. Parallel (a) and Cascade (b) springs
ܭ்௢௧௔௟ ൌ ܭଵ ൅ ܭଶ (2)
ଵ
௄೅೚೟ೌ೗
ൌ ଵ
௄భ
൅ ଵ
௄మ
(3)
The above equations are extracted from length of model and their effective forces on cascade and
parallel system like below (eq.4, 5):
ܨ்௢௧௔௟ ൌ ܨଵ ൌ ܨଶ
ܮ்௢௧௔௟ ൌ ܮଵ ൅ ܮଶ
൜
(4)
ܮ்௢௧௔௟ ൌ ܮଵ ൌ ܮଶ
ܨ்௢௧௔௟ ൌ ܨଵ ൅ ܨଶ
൜
(5)
Now, I'm going to start with these assumptions and derive Maxwell and Kelvin formulations for
viscoelasticity models. At first for Maxwell (Figure 1) we have:
ቄ
ߪ ൌ ߪଵ ൌ ߪଶ
ܮ ൌ ܮଵ ൅ ܮଶ , Δܮ ൌ Δܮଵ ൅ Δܮଶ
(6)
ߝ ൌ Δ௅
௅
(7)

3. So, we can divide part 2 of (eq.6) to Δܮ:
(8)
௅
Δ௅ ൌ ௅భ
Δ௅ ൅ ௅మ
Δ௅
Δ௅ୀΔ௅భାΔ௅మ ሳልልልልልልልሰ ௅
Δ௅ ൌ ௅భ
Δ௅భାΔ௅భ
൅ ௅మ
Δ௅మାΔ௅భ
௜௡௩௘௥௦௘
ሳልልልልሰ Δ௅
௅ ൌ Δ௅భାΔ௅భ
௅భ
൅ Δ௅మାΔ௅భ
௅మ
֜ Δ௅
௅ ൌ Δ௅భ
௅భ
൅ Δ௅మ
௅మ
൅ Δ௅మ
௅భ
൅ Δ௅భ
௅మ
௘௤.଻
ሳልሰ ߝ ൌ ߝଵ ൅ ߝଶ ൅ Δ௅భΔ௅మ
௅బమΔ௅మା௅బభΔ௅భ
Now, if we consider (eq.9) then
Δ௅భΔ௅మ
௅బమΔ௅మା௅బభΔ௅భ
ൎ 0 (9)
ߝ ൌ ߝଵ ൅ ߝଶ (10)
From (eq. 6, 10) for a cascade model and linear relation between stress, strain and Yang's modulus
(eq. 11, 12) and using (eq.13), we can reach (eq. 14) for (Figure 1) which is Maxwell equation for
viscoelastic materials.
ߪ ൌ ߟߝ (11)
ߪ ൌ ߪଵ ൌ ߟߝଵ , ߪ ൌ ߪଶ ൌ ߟ෤ߝሶଵ
(12)
ߪ ൌ ߟߝ
೏ሺ ሻ
೏೟ ሳሰ ߪሶൌ ߟሶߝ ൅ ߟߝሶ
ఎሶୀ଴ ௙௢௥ ௟௜௡௘௔௥ ௒௔௡௚ᇲ௦ ௠௢ௗ௨௟௨௦
ሳልልልልልልልልልልልልልልልልልልልልልልልሰ ߪሶൌ ߟߝሶ (13)
ߝሶൌ ߝሶଵ
൅ ߝሶଶ
௘௤.ଵଶ,ଵଷ
ሳልልልልሰ ߝሶൌ ఙሶ
ఎ ൅ ఙ
ఎ෥
(14)
In continue for parallel model of viscoelastic material we have below equations:
Δܮ ൌ Δܮଵ ൌ Δܮଶ
ܮ ൌ ܮଵ ൌ ܮଶ ൠ ֜ ߝ ൌ ߝଵ ൌ ߝଶ (15)
൜
By fill (eq.12) in part one of (eq.15) we will have:
ߪ ൌ ߝߟ ൅ ߝሶߟ෤ (16)
This is known as Kelvin equation for viscoelastic materials. From method of derivation of Kelvin
equation and from (eq.9), we can say that the Kelvin bar model doesn't have any vanishing
assumption so in compare to Maxwell model, Kelvin model has better accuracy in matching with real
behaviors of viscoelastic material.
Creep and Relaxation
Creep is a kind of phenomena that we have strain of specimen under constant stress during time, so
mathematically it happens when ߪሶൌ 0, and in Relaxation we face with constant strain respect to

4. time (ߝሶൌ 0), so we can extract equation of stress during time for Relaxation phenomena of a
viscoelastic material from Maxwell equation (eq.17).
ߪሶൌ െ ఎ
ఎ෥ ߪ (17)
Now we can separate variables and solve ODE1 like below:
ௗఙ
ఙ ൌ െ ఎ
ఎ෥ ݀ݐ (18)
ߪ ൌ ߪ଴݁ିആ
ആ෥௧
(19)
The above equation is the relation for Relaxation test on a Maxwell bar.
Figure 3. Relaxation test of a Maxwell bar
Conclusion
Briefly, from deriving Maxwell and Kelvin equations for viscoelastic materials, we could find that the
Kelvin model will have more similar results with real model in fast deformations which we couldn't
vanish some parts.
Next duty
I'm going to find different definitions of crack in materials and compare them.
1 . Ordinary Differential Equation

5. References
[1]. "Continuum Mechanics", Fridtjov Irgens, © 2008 Springer‐Verlag Berlin Heidelberg, e‐ISBN 978‐
3‐0‐74289‐2