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Dynamic modulus

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Dynamic modulus (sometimes complex modulus^[1]) is the ratio of stress to strain under vibratory conditions (calculated from data obtained from either free or forced vibration tests, in shear, compression, or elongation). It is a property of viscoelastic materials.

Viscoelastic stress–strain phase-lag

[edit]

Viscoelasticity is studied using dynamic mechanical analysis where an oscillatory force (stress) is applied to a material and the resulting displacement (strain) is measured.^[2]

In purely elastic materials the stress and strain occur in phase, so that the response of one occurs simultaneously with the other.
In purely viscous materials, there is a phase difference between stress and strain, where strain lags stress by a 90 degree ( $\pi /2$ radian) phase lag.
Viscoelastic materials exhibit behavior somewhere in between that of purely viscous and purely elastic materials, exhibiting some phase lag in strain.^[3]

Stress and strain in a viscoelastic material can be represented using the following expressions:

Strain: $\varepsilon =\varepsilon _{0}\sin(\omega t)$
Stress: $\sigma =\sigma _{0}\sin(\omega t+\delta )\,$ ^[3]

where

\omega =2\pi f

where

f

is frequency of strain oscillation,

t

is time,

\delta

is phase lag between stress and strain.

The stress relaxation modulus $G\left(t\right)$ is the ratio of the stress remaining at time $t$ after a step strain $\varepsilon$ was applied at time $t=0$ : $G\left(t\right)={\frac {\sigma \left(t\right)}{\varepsilon }}$ ,

which is the time-dependent generalization of Hooke's law. For visco-elastic solids, $G\left(t\right)$ converges to the equilibrium shear modulus^[4] $G$ :

G=\lim _{t\to \infty }G(t)

The Fourier transform of the shear relaxation modulus $G(t)$ is ${\hat {G}}(\omega )={\hat {G}}'(\omega )+i{\hat {G}}''(\omega )$ (see below).

Storage and loss modulus

[edit]

The storage and loss modulus in viscoelastic materials measure the stored energy, representing the elastic portion, and the energy dissipated as heat, representing the viscous portion.^[3] The tensile storage and loss moduli are defined as follows:

Storage: $E'={\frac {\sigma _{0}}{\varepsilon _{0}}}\cos \delta$
Loss: $E''={\frac {\sigma _{0}}{\varepsilon _{0}}}\sin \delta$ ^[3]

Similarly we also define shear storage and shear loss moduli, $G'$ and $G''$ .

Complex variables can be used to express the moduli $E^{*}$ and $G^{*}$ as follows:

E^{*}=E'+iE''\,

G^{*}=G'+iG''\,

^[3]

where $i$ is the imaginary unit.

Ratio between loss and storage modulus

[edit]

The ratio of the loss modulus to storage modulus in a viscoelastic material is defined as the $\tan \delta$ , (cf. loss tangent), which provides a measure of damping in the material. $\tan \delta$ can also be visualized as the tangent of the phase angle ( $\delta$ ) between the storage and loss modulus.

Tensile: $\tan \delta ={\frac {E''}{E'}}$

Shear: $\tan \delta ={\frac {G''}{G'}}$

For a material with a $\tan \delta$ greater than 1, the energy-dissipating, viscous component of the complex modulus prevails.

References

[edit]

^ The Open University (UK), 2000. T838 Design and Manufacture with Polymers: Solid properties and design, page 30. Milton Keynes: The Open University.
^ "PerkinElmer "Mechanical Properties of Films and Coatings"" (PDF). Archived from the original (PDF) on 2008-09-16. Retrieved 2009-05-09.
^ ^a ^b ^c ^d ^e Meyers and Chawla (1999): "Mechanical Behavior of Materials," 98-103.
^ Rubinstein, Michael, 1956 December 20- (2003). Polymer physics. Colby, Ralph H. Oxford: Oxford University Press. p. 284. ISBN 019852059X. OCLC 50339757.{{cite book}}: CS1 maint: multiple names: authors list (link) CS1 maint: numeric names: authors list (link)

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The dynamic modulus, often denoted as

E^*

, is a complex-valued property that characterizes the stress-strain relationship in linear viscoelastic materials subjected to sinusoidal or oscillatory loading, capturing both elastic and viscous responses under dynamic conditions. It is defined such that its magnitude |

E^*

| is the ratio of the peak stress amplitude to the peak strain amplitude, |

E^*

| =

\sigma_0 / \epsilon_0

, and serves as a fundamental measure of material stiffness in time-dependent deformation scenarios.^[1]^[2] The dynamic modulus comprises two key components: the storage modulus

E'

, which is the real part and quantifies the elastic energy stored and recovered during each cycle of deformation, and the loss modulus

E''

, which is the imaginary part and represents the energy dissipated as heat due to viscous damping.^[1] The magnitude

|E^*|

provides the overall stiffness, while the phase angle

\delta

(where

\tan \delta = E'' / E'

) indicates the relative contributions of elastic and viscous behaviors, with higher

\delta

values signifying greater viscous dominance.^[2] These components are frequency- and temperature-dependent, reflecting the material's relaxation and retardation processes in viscoelastic models like the Maxwell or Kelvin-Voigt elements.^[3] Dynamic modulus is commonly measured using techniques such as dynamic mechanical analysis (DMA), which applies controlled sinusoidal strains or stresses over a range of frequencies (typically 0.01 Hz to 100 Hz) and temperatures to generate master curves via time-temperature superposition principles.^[3] In applications, it is essential for characterizing materials like polymers, asphalt concrete mixtures, and biological tissues, informing design in fields such as pavement engineering, where it predicts rutting and fatigue under traffic loads, and polymer processing, where it guides formulation for optimal mechanical performance.^[2]^[1]^[4] For instance, in hot-mix asphalt, dynamic modulus values decrease with increasing temperature and rise with loading frequency, influencing mechanistic-empirical pavement design.^[2]

Fundamentals

Definition

The dynamic modulus

E^*

is a complex-valued measure of a material's mechanical response to oscillatory or sinusoidal loading, characterized by the ratio of the complex stress amplitude to the complex strain amplitude, which accounts for phase differences between stress and strain waveforms. Its magnitude is given by

|E^*| = \frac{\sigma_0}{\epsilon_0}

, where

\sigma_0

represents the peak stress amplitude and

\epsilon_0

the peak strain amplitude under linear viscoelastic conditions.^[5] This characterization is particularly relevant for materials subjected to cyclic deformations, such as in dynamic mechanical analysis (DMA).^[6] Unlike the static modulus, which quantifies material stiffness under steady-state loading and neglects time-dependent behaviors like creep or relaxation, the dynamic modulus captures the viscoelastic nature of materials that exhibit both elastic recovery and viscous dissipation during oscillation.^[6] Static tests, such as those following Hooke's law with constant strain, provide a time-independent elastic constant, whereas dynamic testing reveals the material's frequency-sensitive response.^[6] The value of the dynamic modulus depends on the loading frequency and temperature, as these factors influence the balance between elastic and viscous contributions in viscoelastic materials, which display both instantaneous elastic deformation and time-delayed viscous flow.^[7]^[5] For instance, higher frequencies typically increase the modulus by promoting elastic-like behavior, while elevated temperatures enhance viscous effects and reduce overall stiffness.^[7]

Relation to viscoelasticity

Viscoelasticity describes the mechanical behavior of materials that exhibit both viscous and elastic characteristics under deformation, where the elastic component enables energy storage and reversible deformation akin to a spring, while the viscous component results in energy dissipation and time-dependent flow similar to a fluid.^[6] This dual nature arises because the stress-strain relationship in such materials depends on the rate and duration of loading, leading to phenomena like creep (continued deformation under constant stress) and stress relaxation (decreasing stress under constant strain).^[1] Dynamic modulus quantifies the viscoelastic response under cyclic loading by capturing how materials respond to oscillatory forces.^[8] Foundational models for understanding the dynamic response of viscoelastic materials include the Maxwell model and the Kelvin-Voigt model, which idealize these behaviors using combinations of springs (elastic elements) and dashpots (viscous elements).^[1] The Maxwell model, consisting of a spring and dashpot in series, represents materials that show fluid-like flow over long timescales, such as polymer melts, and is useful for analyzing relaxation processes that influence dynamic properties under vibration.^[9] In contrast, the Kelvin-Voigt model, with a spring and dashpot in parallel, models solid-like materials that resist instantaneous deformation but exhibit delayed recovery, providing insight into creep and the time-lagged response in oscillatory conditions typical of rubbers and gels.^[10] These models serve as building blocks for more complex representations of dynamic viscoelasticity, highlighting how internal friction and elasticity interact during periodic loading.^[1] Viscoelastic analysis can be conducted in the time domain, focusing on transient responses like step loading, or in the frequency domain, which examines steady-state behavior under harmonic oscillations where material properties vary with oscillation frequency.^[6] In the frequency domain, dynamic modulus applies specifically to sinusoidal inputs, revealing how viscous dissipation increases at higher frequencies while elastic storage dominates at lower ones, a shift critical for materials subjected to vibrations.^[8] This approach contrasts with time-domain methods by leveraging Fourier transforms to interconvert responses, enabling the study of periodic deformations without solving full transient equations.^[8] The concept of dynamic modulus emerged in mid-20th century research on polymers and rubbers, driven by efforts to characterize their time-dependent behaviors under practical loading conditions.^[11] Early foundational work, such as that by Herbert Leaderman in the 1940s, focused on creep and recovery in high polymers like filaments and plastics, laying the groundwork for understanding oscillatory responses in these materials.^[12] Leaderman's investigations, detailed in his 1943 monograph on elastic and creep properties, highlighted the need for frequency-dependent measures to model real-world applications in rubber elasticity and polymer engineering.

Mathematical Formulation

Complex modulus

The complex modulus, denoted as

E^*(\omega)

, is a fundamental quantity in the frequency-domain analysis of viscoelastic materials, expressed as

E^*(\omega) = E' + i E''

, where

E'

is the real part,

E''

is the imaginary part, and

\omega

is the angular frequency.^[9]^[1] This representation arises from applying the Fourier transform to the time-dependent stress

\sigma(t)

and strain

\epsilon(t)

, transforming the convolution integral of the linear viscoelastic constitutive equation into a simple algebraic relation in the frequency domain:

\hat{\sigma}(\omega) = E^*(\omega) \hat{\epsilon}(\omega)

, where

\hat{\sigma}(\omega)

and

\hat{\epsilon}(\omega)

are the Fourier transforms of stress and strain, respectively.^[13] For harmonic oscillations, such as

\epsilon(t) = \epsilon_0 \cos(\omega t)

, the corresponding stress is

\sigma(t) = \sigma_0 \cos(\omega t + \delta)

, leading to

E^*(\omega) = \frac{\sigma_0}{\epsilon_0} e^{i \delta}

.^[1] The magnitude of the complex modulus,

|E^*| = \sqrt{(E')^2 + (E'')^2}

∣E∗∣=(E′)2+(E′′)2

, quantifies the overall stiffness under dynamic loading, while the phase angle $\delta$ captures the lag between stress and strain, given by $\tan \delta = \frac{E''}{E'}$ .^[9]^[14] This phase relation links directly to energy dissipation, as $\sin \delta$ represents the fraction of input energy lost per cycle.^[13] The complex modulus $E^*(\omega)$ exhibits strong frequency dependence, reflecting the material's transition from viscous- to elastic-dominated behavior.^[1] At low frequencies ( $\omega \to 0$ ), $E^*(\omega)$ approaches the static equilibrium modulus, dominated by long-term relaxation processes.^[13] Conversely, at high frequencies ( $\omega \to \infty$ ), it approaches the instantaneous glassy modulus, where the material responds as a rigid elastic solid.^[9] The storage modulus $E'$ represents the elastic energy storage, while the loss modulus $E''$ accounts for viscous dissipation.^[1]

Storage and loss components

The storage modulus, denoted as

E'

, quantifies the elastic component of a viscoelastic material's response under oscillatory loading, representing the energy stored and recoverable during deformation, much like the Young's modulus in purely elastic solids.^[1] This component reflects the material's stiffness and ability to store mechanical energy as potential energy without permanent deformation.^[4] In contrast, the loss modulus,

E''

, captures the viscous dissipation of energy as heat due to internal friction and molecular rearrangements during cyclic straining.^[15] Together, these real and imaginary parts form the complex dynamic modulus

E^*

, which encapsulates the overall viscoelastic behavior.^[16] The storage and loss moduli are derived from the magnitude

|E^*|

of the complex modulus and the phase angle

\delta

between stress and strain, with the following relationships:

E' = |E^*| \cos \delta

E'' = |E^*| \sin \delta

These expressions highlight how

E'

dominates at low

\delta

(predominantly elastic response) and

E''

at high

\delta

(predominantly viscous response).^[17] In glassy polymers below their glass transition temperature, the storage modulus

E'

typically increases with applied frequency, as higher rates of deformation restrict chain mobility and enhance the elastic character.^[18] Conversely, the loss modulus

E''

exhibits a peak near the glass transition temperature, where molecular relaxations are most pronounced, leading to maximum energy dissipation.^[19] The ratio

E'' / E'

serves as a measure of the material's damping capacity, indicating its effectiveness in absorbing vibrational energy without excessive rebound.^[20]

Viscoelastic Response

Phase lag in stress-strain

In viscoelastic materials subjected to oscillatory loading, the phase lag δ represents the angular difference by which the strain response lags behind the applied stress, quantifying the material's combined elastic and viscous behavior. For an ideally elastic material, δ = 0°, where stress and strain are perfectly in phase, indicating no energy dissipation. In contrast, for a purely viscous material, δ = 90°, with strain maximally delayed relative to stress, reflecting complete energy dissipation through flow. Viscoelastic materials exhibit intermediate values of δ between 0° and 90°, capturing the time-dependent interplay of recoverable deformation and internal friction.^[21]^[22]^[23] Under sinusoidal oscillatory conditions, the stress is typically expressed as σ(t) = σ₀ cos(ωt), where σ₀ is the stress amplitude and ω is the angular frequency, while the resulting strain follows ε(t) = ε₀ cos(ωt - δ), with ε₀ as the strain amplitude. This phase shift δ arises from the material's relaxation processes, where molecular rearrangements delay the strain response. In dynamic mechanical testing, δ serves as a direct measure of hysteresis, characterizing the material's ability to store and release energy versus dissipate it as heat.^[1]^[23]^[6] The phase lag manifests in the stress-strain response as a closed hysteresis loop over each oscillation cycle, where the area enclosed by the loop corresponds to the energy dissipated per cycle, proportional to π σ₀ ε₀ sin δ. This dissipation stems from frictional losses within the material's microstructure, converting mechanical work into thermal energy. In dynamic testing, hysteresis quantified by δ is essential for assessing damping properties in applications like vibration control.^[1]^[23] The value of δ exhibits strong dependence on the loading frequency ω, reflecting the material's relaxation spectrum—a distribution of relaxation times H(τ) associated with molecular motions. At high frequencies (short timescales), δ approaches 0°, as the material behaves more elastically, with rapid deformations outpacing viscous flow. Conversely, at low frequencies (long timescales), δ increases toward 90°, emphasizing viscous dominance. The frequency variation of δ often peaks in the transition region where elastic and viscous contributions balance, with the shape and position of this peak governed by the breadth and location of the relaxation spectrum; broader spectra, as in entangled polymers, lead to more gradual changes in δ across frequencies. This frequency-dependent behavior, detailed in foundational analyses of polymer dynamics, enables characterization of relaxation processes through dynamic measurements.^[6]^[23]

Loss tangent

The loss tangent, denoted as

\tan \delta

, is defined as the ratio of the loss modulus

E''

to the storage modulus

E'

, i.e.,

\tan \delta = E'' / E'

.^[23] This ratio measures the relative energy dissipation to energy storage in a viscoelastic material under oscillatory loading. It arises from the phase lag between stress and strain in the viscoelastic response. Mathematically,

\tan \delta = \sin \delta / \cos \delta

, where

\delta

is the phase angle, directly linking the parameter to the material's damping behavior.^[23] A high value of

\tan \delta

indicates viscous dominance, where energy dissipation prevails, as seen in rubbers such as butyl rubber (

\tan \delta \approx 0.4

) and neoprene (

\tan \delta \approx 0.1

).^[24]^[23] Conversely, low

\tan \delta

values characterize stiff materials with elastic dominance, such as entangled polymers in the plateau zone or glassy states, where

\tan \delta

can drop below 0.03.^[23] The loss tangent exhibits strong dependence on frequency and temperature, often peaking at molecular transitions like the glass-rubber transition due to enhanced chain mobility and relaxation processes.^[23] At low frequencies,

\tan \delta

may scale inversely with frequency (

\propto \omega^{-1}

) for uncross-linked polymers, while temperature shifts these peaks, analyzable via time-temperature superposition principles.^[23] In applications such as vibration control, materials with

\tan \delta > 0.1

are desirable for energy absorbers, as they effectively convert mechanical vibrations into heat, reducing resonance in structures like machinery and vehicles.^[24] This damping role extends to noise abatement and polymer design for enhanced performance in dynamic environments.^[23]

Measurement and Applications

Experimental techniques

Dynamic Mechanical Analysis (DMA) is the primary experimental technique for measuring the dynamic modulus of viscoelastic solids, such as polymers and composites. In DMA, a sinusoidal strain is applied to the sample using a drive motor, and the resulting stress response is measured by a force transducer, allowing computation of the complex modulus

E^*

, storage modulus

E'

, loss modulus

E''

, and loss tangent

\tan \delta

from the phase difference and amplitudes.^[22]^[17]^[6] This method operates under controlled oscillatory conditions, typically in tension, compression, shear, or bending modes, to characterize material behavior across a range of deformation frequencies and temperatures.^[25] Common DMA protocols include frequency sweeps and temperature ramps to probe the frequency and temperature dependence of the dynamic modulus. Frequency sweeps generally cover ranges from 0.01 Hz to 100 Hz, revealing how moduli vary with oscillation rate, while temperature ramps span -100°C to 200°C to capture transitions like the glass-rubber state in polymers.^[26]^[22] These experiments often use small strain amplitudes (0.01-1%) to remain in the linear viscoelastic regime, ensuring accurate representation of material properties without nonlinear effects.^[6] Recent advancements as of 2025 include the integration of humidity and UV exposure options in instruments like the NETZSCH DMA 303 Eplexor, enabling testing under simulated environmental conditions, and the development of miniaturization for portable DMAs to facilitate on-site measurements.^[27]^[28] Additionally, DMA-inspired devices for in vivo characterization of viscoelastic properties in living tissues, such as human skin, have emerged, allowing non-invasive assessment under physiological conditions.^[29] For liquids and soft materials, rotational rheometers employing oscillatory shear are used to determine the dynamic shear modulus

G^*

, from which the dynamic modulus can be derived assuming isotropy.^[30]^[31] In these setups, a controlled torque applies sinusoidal deformation between parallel plates or cones, measuring the torque response to compute storage and loss shear moduli over frequencies up to 100 rad/s.^[32] Ultrasonic techniques, suitable for solids, involve propagating high-frequency waves (typically 1-10 MHz) through the sample and measuring wave velocities to calculate the dynamic Young's modulus from

E = \rho v^2

, where

\rho

is density and

v

is velocity; this non-destructive method is particularly effective for homogeneous materials like metals or ceramics.^[33]^[34] DMA and related data are often presented as master curves to extend the frequency or time scale beyond experimental limits, constructed via time-temperature superposition (TTS). In TTS, isotherms of modulus versus frequency at different temperatures are horizontally shifted along a logarithmic frequency axis using shift factors

a_T

, typically fitted to the Williams-Landel-Ferry (WLF) equation:

\log a_T = -\frac{C_1 (T - T_0)}{C_2 + (T - T_0)}

, where

T_0

is a reference temperature, and

C_1

C_2

are material constants near the glass transition.^[35]^[26] This approach yields a continuous representation of viscoelastic behavior, valid for amorphous polymers above the glass transition.^[36] Accurate dynamic modulus measurements require attention to artifacts from sample geometry and instrument calibration. Sample dimensions directly influence calculated moduli, as

E' = K \cdot L / A

where

K

is measured stiffness,

L

length, and

A

cross-section; inconsistencies in geometry can introduce errors up to 20% if not precisely controlled.^[37] Calibration of force transducers, displacement sensors, and temperature controls is essential to minimize systematic offsets, often verified using standard materials like polycarbonate before testing.^[38] Poor adhesion or slippage in fixtures can also artifactually lower apparent moduli, necessitating surface treatments or clamps.^[39]

Practical uses in materials science

In materials science, dynamic modulus plays a crucial role in material selection by enabling engineers to evaluate stiffness and energy dissipation for specific performance requirements. For structural composites, a high storage modulus (E') indicates superior load-bearing capacity under oscillatory conditions, guiding the choice of fiber-reinforced polymers for aerospace and automotive components where rigidity is paramount. Conversely, impact-resistant polymers, such as toughened epoxies, benefit from a balanced loss modulus (E'') that promotes energy absorption without excessive damping, optimizing toughness in applications like protective gear.^[40]^[41] Quality control processes leverage dynamic modulus evolution to monitor curing in adhesives and coatings, ensuring consistent mechanical integrity during production. During thermoset curing, the transition from low to high modulus reflects gelation and vitrification stages, allowing real-time assessment of cure completeness and defect detection in bonded assemblies for electronics and aerospace. This approach provides quantitative metrics, such as modulus increase rates, to validate process parameters and batch uniformity without destructive testing.^[42] Engineering applications highlight dynamic modulus in optimizing component performance under cyclic loading. In automotive tires, the loss tangent (tan δ) at elevated temperatures correlates with rolling resistance, where low values minimize fuel consumption by reducing hysteresis losses in rubber compounds during repeated deformation. For biomedical implants, such as silicone-based prosthetics, dynamic modulus assesses fatigue resistance under physiological cyclic stresses, predicting long-term durability and biocompatibility by quantifying viscoelastic degradation over millions of cycles.^[43]^[44] Modern advancements extend dynamic modulus to emerging fields like 3D printing resins and nanocomposites, where it predicts interlayer adhesion and overall structural integrity. In stereolithography resins, frequency-dependent modulus measurements inform print parameters to enhance layer bonding, reducing delamination in functional prototypes. For nanocomposites, such as carbon nanotube-filled polymers, dynamic analysis reveals reinforcement effects on modulus enhancement, enabling tailored designs for lightweight, high-strength applications in electronics and energy storage. Recent applications as of 2025 include DMA for organic semiconductors to understand thermal transitions and viscoelastic properties in flexible electronics, and for hydrogels in biomedical scaffolds using combined axial-torsional testing.^[45]^[46]^[47]^[48] Despite its versatility, dynamic modulus has limitations in materials science, as it primarily captures viscoelastic responses and is less effective for purely plastic deformation regimes where permanent yielding dominates. It complements static tensile tests by providing frequency- and temperature-resolved insights but requires integration with other methods for comprehensive characterization of nonlinear behaviors.^[22]

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Dynamic modulus

Fundamentals

Definition

Relation to viscoelasticity

Mathematical Formulation

Complex modulus

Storage and loss components

Viscoelastic Response

Phase lag in stress-strain

Loss tangent

Measurement and Applications

Experimental techniques

Practical uses in materials science

References

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Mathematical Formulation

Complex modulus

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