12 KiB
| title | chunk | source | category | tags | date_saved | instance |
|---|---|---|---|---|---|---|
| Dynamic mechanical analysis | 1/3 | https://en.wikipedia.org/wiki/Dynamic_mechanical_analysis | reference | science, encyclopedia | 2026-05-05T10:04:19.735755+00:00 | kb-cron |
Dynamic mechanical analysis (abbreviated DMA) is a technique used to study and characterize materials. It is most useful for studying the viscoelastic behavior of polymers. A sinusoidal stress is applied and the strain in the material is measured, allowing one to determine the complex modulus. The temperature of the sample or the frequency of the stress are often varied, leading to variations in the complex modulus; this approach can be used to locate the glass transition temperature of the material, as well as to identify transitions corresponding to other molecular motions.
== Theory ==
=== Viscoelastic properties of materials ===
Polymers composed of long molecular chains have unique viscoelastic properties, which combine the characteristics of elastic solids and Newtonian fluids. The classical theory of elasticity describes the mechanical properties of elastic solids where stress is proportional to strain in small deformations. Such response to stress is independent of strain rate. The classical theory of hydrodynamics describes the properties of viscous fluid, for which stress response depends on strain rate. This solidlike and liquidlike behaviour of polymers can be modelled mechanically with combinations of springs and dashpots, making for both elastic and viscous behaviour of viscoelastic materials such as bitumen.
=== Dynamic moduli of polymers === The viscoelastic property of a polymer is studied by dynamic mechanical analysis where a sinusoidal force (stress σ) is applied to a material and the resulting displacement (strain) is measured. For a perfectly elastic solid, the resulting strain and the stress will be perfectly in phase. For a purely viscous fluid, there will be a 90 degree phase lag of strain with respect to stress. Viscoelastic polymers have the characteristics in between where some phase lag will occur during DMA tests. When the strain is applied and the stress lags behind, the following equations hold:
Stress:
σ
=
σ
0
sin
(
t
ω
+
δ
)
{\displaystyle \sigma =\sigma _{0}\sin(t\omega +\delta )\,}
Strain:
ε
=
ε
0
sin
(
t
ω
)
{\displaystyle \varepsilon =\varepsilon _{0}\sin(t\omega )}
where
ω
{\displaystyle \omega }
is the frequency of strain oscillation,
t
{\displaystyle t}
is time,
δ
{\displaystyle \delta }
is phase lag between stress and strain. Consider the purely elastic case, where stress is proportional to strain given by Young's modulus
E
{\displaystyle E}
. We have
σ
(
t
)
=
E
ϵ
(
t
)
⟹
σ
0
sin
(
ω
t
+
δ
)
=
E
ϵ
0
sin
ω
t
⟹
δ
=
0
{\displaystyle \sigma (t)=E\epsilon (t)\implies \sigma _{0}\sin {(\omega t+\delta )}=E\epsilon _{0}\sin {\omega t}\implies \delta =0}
Now for the purely viscous case, where stress is proportional to strain rate.
σ
(
t
)
=
K
d
ϵ
d
t
⟹
σ
0
sin
(
ω
t
+
δ
)
=
K
ϵ
0
ω
cos
ω
t
⟹
δ
=
π
2
{\displaystyle \sigma (t)=K{\frac {d\epsilon }{dt}}\implies \sigma _{0}\sin {(\omega t+\delta )}=K\epsilon _{0}\omega \cos {\omega t}\implies \delta ={\frac {\pi }{2}}}
The storage modulus measures the stored energy, representing the elastic portion, and the loss modulus measures the energy dissipated as heat, representing the viscous portion. The tensile storage and loss moduli are defined as follows:
Storage modulus:
E
′
=
σ
0
ε
0
cos
δ
{\displaystyle E'={\frac {\sigma _{0}}{\varepsilon _{0}}}\cos \delta }
Loss modulus:
E
″
=
σ
0
ε
0
sin
δ
{\displaystyle E''={\frac {\sigma _{0}}{\varepsilon _{0}}}\sin \delta }
Phase angle:
δ
=
arctan
E
″
E
′
{\displaystyle \delta =\arctan {\frac {E''}{E'}}}
Similarly, in the shearing instead of tension case, we also define shear storage and loss moduli,
G
′
{\displaystyle G'}
and
G
″
{\displaystyle G''}
. Complex variables can be used to express the moduli
E
∗
{\displaystyle E^{*}}
and
G
∗
{\displaystyle G^{*}}
as follows:
E
∗
=
E
′
+
i
E
″
=
σ
0
ε
0
e
i
δ
{\displaystyle E^{*}=E'+iE''={\frac {\sigma _{0}}{\varepsilon _{0}}}e^{i\delta }\,}
G
∗
=
G
′
+
i
G
″
{\displaystyle G^{*}=G'+iG''\,}
where
i
2
=
−
1
{\displaystyle {i}^{2}=-1\,}
==== Derivation of dynamic moduli ==== Shear stress
σ
(
t
)
=
∫
−
∞
t
G
(
t
−
t
′
)
γ
˙
(
t
′
)
d
t
′
{\displaystyle \sigma (t)=\int _{-\infty }^{t}G(t-t'){\dot {\gamma }}(t')dt'}
of a finite element in one direction can be expressed with relaxation modulus
G
(
t
−
t
′
)
{\displaystyle G(t-t')}
and strain rate, integrated over all past times
t
′
{\displaystyle t'}
up to the current time
t
{\displaystyle t}
. With strain rate
γ
(
t
)
˙
=
ω
⋅
γ
0
⋅
cos
(
ω
t
)
{\displaystyle {\dot {\gamma (t)}}=\omega \cdot \gamma _{0}\cdot \cos(\omega t)}
and substitution
ξ
(
t
′
)
=
t
−
t
′
=
s
{\displaystyle \xi (t')=t-t'=s}
one obtains
σ
(
t
)
=
∫
ξ
(
−
∞
)
=
t
−
(
−
∞
)
ξ
(
t
)
=
t
−
t
G
(
s
)
ω
γ
0
⋅
cos
(
ω
(
t
−
s
)
)
(
−
d
s
)
=
γ
0
∫
0
∞
ω
G
(
s
)
cos
(
ω
(
t
−
s
)
)
d
s
{\displaystyle \sigma (t)=\int _{\xi (-\infty )=t-(-\infty )}^{\xi (t)=t-t}G(s)\omega \gamma _{0}\cdot \cos(\omega (t-s))(-ds)=\gamma _{0}\int _{0}^{\infty }\omega G(s)\cos(\omega (t-s))ds}
. Application of the trigonometric addition theorem
cos
(
x
±
y
)
=
cos
(
x
)
cos
(
y
)
∓
sin
(
x
)
sin
(
y
)
{\displaystyle \cos(x\pm y)=\cos(x)\cos(y)\mp \sin(x)\sin(y)}
lead to the expression