Fact-checked by Grok 2 weeks ago

Shear modulus

The shear modulus, also known as the modulus of rigidity and denoted by G, is a fundamental measure of a material's elastic stiffness against shear deformation, defined as the ratio of applied shear stress to the resulting shear strain within the linear elastic regime.^[1]^[2] Shear deformation occurs when tangential forces cause adjacent layers of a material to slide relative to one another, such as in torsion or parallel displacement scenarios.^[1] In mathematical terms, the shear modulus relates shear stress τ (force per unit area) to shear strain γ (the angular distortion, often approximated as γ ≈ tan θ for small angles θ) through the Hookean linear relationship τ = G γ, where γ is dimensionless, giving G units of pressure, such as pascals (Pa) in the SI system.^[1]^[2] This constant is interconnected with other elastic properties, including Young's modulus E (which measures axial stiffness) and Poisson's ratio ν (which quantifies lateral contraction under axial load), via the relation G = E / [2(1 + ν)], allowing estimation from tensile test data.^[2] The shear modulus plays a critical role in materials science and engineering, characterizing a material's resistance to shear distortion and enabling predictions of behavior under loads like torsion in shafts, beams, or structural components.^[3]^[4] Values vary widely by material—for instance, metals like steel exhibit high G around 80 GPa, indicating rigidity, while softer materials like rubber have much lower values near 0.01 GPa, reflecting flexibility—making it essential for selecting materials in applications from aerospace structures to geotechnical analysis.^[2] It is typically determined experimentally through torsion tests or dynamic methods, ensuring accurate modeling of elastic limits before yielding or plastic deformation occurs.^[2]

Fundamentals

Definition

The shear modulus, denoted by G, is defined as the ratio of shear stress \tau to shear strain \gamma for a material undergoing simple shear deformation, expressed mathematically as

G = \frac{\tau}{\gamma}.

This constant quantifies the material's resistance to shearing forces that cause layers to slide parallel to each other./20%3A_Miscellaneous/20.03%3A_Shear_Modulus_and_Torsion_Constant) Shear deformation represents a distortion of the material's shape without altering its volume, as the trace of the infinitesimal strain tensor is zero in pure shear, distinguishing it from tensile or compressive deformations that primarily involve changes in length and potentially volume.^[5] In the International System of Units (SI), the shear modulus is measured in pascals (Pa), equivalent to newtons per square meter (N/m²), though values for engineering materials are often reported in gigapascals (GPa) due to their magnitude./20%3A_Miscellaneous/20.03%3A_Shear_Modulus_and_Torsion_Constant) The concept was introduced by Thomas Young in 1807 as part of his work on elastic constants in "A Course of Lectures on Natural Philosophy and the Mechanical Arts," where he described shear deformation and its associated modulus.^[6] Early experimental measurements of the shear modulus were conducted by Guillaume Wertheim in 1848 through studies on the elasticity of homogeneous solid bodies.^[7] For homogeneous and isotropic materials, where properties are uniform and independent of direction, the shear modulus is a scalar quantity. In anisotropic materials, such as crystals or composites, it takes the form of a fourth-rank tensor to account for direction-dependent stiffness.^[3]

Relation to Other Moduli

In isotropic linear elastic materials, the shear modulus G is interrelated with other elastic constants, such as Young's modulus E and Poisson's ratio \nu, through the equation G = \frac{E}{2(1 + \nu)}. This relation arises from the equivalence of strain energy under uniaxial tension and pure shear deformation, ensuring consistency in the Hookean constitutive laws for isotropic solids.^[8] Similarly, G connects to the bulk modulus K via G = \frac{3K(1 - 2\nu)}{2(1 + \nu)}, which derives from the volumetric response under hydrostatic stress and the incompressibility limit approached as \nu \to 0.5.^[9] These interdependencies extend to the Lamé parameters, where the shear modulus corresponds directly to the second Lamé constant \mu = G, while the first Lamé constant \lambda relates to the bulk modulus through K = \lambda + \frac{2}{3}\mu. This formulation parameterizes the isotropic elasticity tensor compactly, with \lambda capturing the coupling between normal strains and the resistance to uniform compression.^[10] For thermodynamic stability in isotropic materials, the elastic energy must be positive definite, constraining Poisson's ratio to -1 < \nu < 0.5, which in turn requires G > 0 to prevent unphysical negative stiffness.^[11] As an illustrative example, for typical structural steel with E \approx 200 GPa and \nu \approx 0.3, the shear modulus computes to G \approx 77 GPa using the relation with Young's modulus. In anisotropic materials, such as cubic crystals, the shear modulus is not a single scalar but involves direction-dependent components; notably, the stiffness constant C_{44} represents a key shear modulus for specific orientations, like shear in the {100} plane.^[12]

Physical Properties and Behavior

Shear Waves and Propagation

Shear waves, also known as S-waves, are transverse elastic waves that propagate through solids, characterized by particle displacements perpendicular to the direction of wave propagation. Unlike longitudinal waves, these transverse motions induce shear stresses in the medium, making the shear modulus a key parameter governing their behavior. This perpendicular motion distinguishes S-waves from other seismic or acoustic waves and underscores their sensitivity to the material's resistance to shearing deformation.^[13] The propagation speed of shear waves in an isotropic elastic solid is determined by the formula

v_s = \sqrt{\frac{G}{\rho}}

where G is the shear modulus and \rho is the material density. This expression arises from applying Newton's second law to the harmonic oscillation of particles under shear stress, coupled with Hooke's law relating shear stress to strain via the shear modulus; the resulting wave equation yields the speed as the square root of the ratio of elastic stiffness to inertial density. In comparison, longitudinal P-waves travel at

v_p = \sqrt{\frac{K + \frac{4}{3} G}{\rho}}

where K is the bulk modulus, illustrating how the shear modulus contributes to overall rigidity but is solely responsible for S-wave propagation, as P-waves rely more on compressional resistance. These speeds reflect the fundamental interplay between elastic properties and density in wave dynamics.^[13] In solid media, shear waves exhibit polarization depending on the orientation of particle motion relative to the propagation direction and the surface. Horizontally polarized shear waves (SH waves) feature particle motion parallel to the free surface and perpendicular to propagation, while vertically polarized shear waves (SV waves) involve motion in the vertical plane containing the propagation direction. This polarization allows SH and SV components to respond differently to material anisotropy and interfaces, aiding in detailed imaging of subsurface structures.^[14] Seismological applications highlight the diagnostic value of shear waves, as their inability to traverse fluids—where the shear modulus G = 0—creates distinct shadow zones in earthquake records. For instance, S-waves halt at the core-mantle boundary, producing an S-wave shadow zone opposite the epicenter and providing evidence for Earth's liquid outer core. This property, rooted in the absence of shear rigidity in fluids, contrasts with P-waves and enables differentiation of solid and fluid layers in planetary interiors.^[15]^[16] During propagation, shear waves undergo attenuation primarily due to internal friction, where energy is dissipated as heat through irreversible microscopic processes like dislocation motion or grain boundary sliding. This damping reduces wave amplitude over distance without altering the ideal elastic speed formula, though it introduces frequency-dependent losses in real materials. Such attenuation provides insights into material microstructure and energy dissipation mechanisms in geophysical contexts.^[17]

Temperature and Frequency Dependence

The shear modulus of elastic materials decreases with increasing temperature primarily due to thermal expansion, which widens interatomic spacings and reduces bonding strength, and phonon softening, whereby thermal agitation introduces anharmonicity in lattice vibrations, thereby weakening the effective interatomic potentials.^[18]^[19] Near the melting point, this softening intensifies, causing the shear modulus to approach zero as the solid loses its capacity to sustain shear stresses, foreshadowing the zero shear rigidity of the liquid phase.^[20] Typical temperature dependence curves for metals, such as those derived from ultrasonic measurements, show a roughly linear decline in shear modulus at low temperatures (up to about 0.5 Tm, where Tm is the melting temperature) followed by a steeper drop near Tm, reflecting enhanced anharmonic effects.^[21] For instance, in aluminum, the shear modulus decreases from approximately 26 GPa at room temperature to about 18 GPa at 500°C, a reduction of roughly 31% over this range, as measured using piezoelectric ultrasonic composite oscillator techniques up to near the melting point.^[22] In the ideal elastic regime, applicable to most crystalline solids at ambient conditions, the shear modulus remains independent of frequency, as deformation responses are instantaneous and non-dissipative.^[23] However, at sufficiently high frequencies—typically in the ultrasonic or hypersonic range—or in materials approaching viscoelastic transitions, the modulus becomes frequency-dependent, with the storage modulus increasing and approaching the low-frequency elastic limit while energy dissipation rises.^[24] Anelasticity, stemming from internal friction processes like localized atomic rearrangements or defect interactions, introduces subtle frequency dispersion in the shear modulus, particularly below the glass transition in amorphous solids or at elevated temperatures in crystals, where the real part of the complex modulus shows mild logarithmic variation over seismic to audio frequencies.^[25]^[26]

Measurement and Applications

Experimental Determination

The shear modulus, a key measure of a material's resistance to shear deformation, is determined experimentally through various laboratory techniques that apply controlled stresses or waves to samples and analyze the resulting responses. These methods are essential for characterizing isotropic and anisotropic materials, with selections depending on sample geometry, material type, and desired frequency range. Common approaches include static and dynamic tests, often standardized to ensure reproducibility and accuracy. One primary method is the torsion test, which involves applying a torque to a cylindrical specimen fixed at one end and measuring the resulting angular twist. The shear modulus G is calculated using the relation G = \frac{T L}{J \theta}, where T is the applied torque, L is the gauge length, J is the polar moment of inertia of the cross-section, and \theta is the twist angle in radians. This technique is particularly suitable for metals and structural materials at room temperature, providing static measurements with typical accuracies of 1-5% when proper alignment is maintained.^[27] Ultrasonic pulse-echo methods offer a non-destructive alternative by sending shear waves through the material and measuring their travel time. Transducers generate pulses that reflect off boundaries, allowing determination of shear wave velocity v_s, from which G = \rho v_s^2 is derived, with \rho as the material density. This approach excels for bulk samples and composites, achieving precisions below 1% for homogeneous materials, though it requires corrections for attenuation in highly damping substances.^[28]^[29] Resonant ultrasound spectroscopy (RUS) involves exciting a sample to vibrate at its natural frequencies and fitting the spectrum to theoretical models to extract elastic constants, including the shear modulus. Using contact transducers on regular geometries like cubes or cylinders, RUS determines all independent elastic moduli simultaneously with uncertainties often under 0.1%, making it ideal for precise characterization of single crystals and polycrystals.^[30]^[31] For frequency-dependent behavior, dynamic mechanical analysis (DMA) applies oscillatory shear strains to thin films or bars and measures the storage modulus G', which represents the elastic component of the shear response. Instruments impose sinusoidal deformations in shear mode while varying temperature or frequency, yielding viscoelastic moduli with resolutions of 0.1-1% over ranges from 0.01 Hz to 100 Hz. This method is widely used for polymers and composites to assess dynamic shear properties.^[32]^[33] Experimental challenges include sample preparation, such as ensuring uniform geometry for torsion tests or minimizing surface roughness for ultrasonic coupling, which can introduce errors up to 10% if neglected. Anisotropy requires orientation-specific measurements and tensor corrections, while clamping artifacts in RUS or DMA may cause spurious resonances or damping overestimation. Error sources like instrumental drift or environmental vibrations are mitigated through calibration and isolation, with overall accuracies varying from 0.5% in ideal cases to 5-10% for heterogeneous samples.^[34]^[30] Standardized procedures, such as ASTM E143 for torsion-based shear modulus determination at room temperature, guide testing for structural materials where creep is negligible, emphasizing torque application limits below 50% of yield shear stress.

Engineering and Geophysical Uses

In structural engineering, the shear modulus is essential for calculating torsional rigidity and shear stresses in beams and structural components subjected to twisting loads, such as those in bridges and aircraft wings. The torsional rigidity, expressed as GJ where J is the polar moment of inertia, quantifies a structure's resistance to angular deformation, enabling engineers to design safe margins against failure in applications like prestressed concrete bridge girders under combined shear and torsion.^[35]^[36]^[37] In geophysics, the shear modulus serves as a primary input for seismic inversion techniques to reconstruct subsurface rigidity profiles, with low values signaling potential fault zones or weakened rock layers that facilitate seismic wave scattering and energy dissipation. These inversions, often based on exact Zoeppritz equations within Bayesian frameworks, allow direct estimation of shear modulus variations to delineate brittle reservoirs or fault structures critical for earthquake hazard mapping and hydrocarbon exploration.^[38]^[39]^[40] For the 1989 Loma Prieta earthquake, seismic velocity models indicated a low-rigidity wedge (with shear wave speeds of 3.3–5.8 km/s extending to mid-crustal depths) between the Zayante and San Andreas faults, highlighting inherent crustal weakness that preconditioned the fault for rupture during the magnitude 6.9 event.^[41] In biomechanics, the shear modulus of soft tissues—typically 1–10 kPa—enables modeling of blood vessel compliance, where it predicts radial and tangential deformations under pulsatile flow-induced shear, informing simulations of vascular patency and aneurysm risk.^[42]^[43] The shear modulus is vital for failure prediction, as it helps differentiate ductile yielding from brittle shear rupture; in elasto-plastic analyses of rocks, it defines the elastic regime before activation of the Mohr-Coulomb yield surface, where shear stress exceeds cohesion plus frictional resistance to initiate failure.^[44]^[45] One limitation of the shear modulus is its restriction to infinitesimal, linear elastic strains, neglecting plastic flow or hardening at high deformations, which can underestimate risks in ductile materials or post-failure scenarios.^[46]

Material-Specific Characteristics

In Metals

The shear modulus of polycrystalline metals at room temperature depends on their composition and crystal structure, with typical values for common engineering metals including approximately 80 GPa for carbon steels, 26 GPa for aluminum alloys like 6061, and 48 GPa for pure copper.^[47]^[48] These values reflect the elastic resistance to shear deformation in bulk forms, where grain boundaries average out directional variations.^[49] Face-centered cubic (FCC) metals generally exhibit lower shear moduli than body-centered cubic (BCC) metals, a trend linked to differences in their slip systems and atomic packing density that influence overall stiffness.^[50] For example, aluminum (FCC structure) has a shear modulus of about 26 GPa, significantly lower than that of iron (BCC structure) at around 81 GPa.^[48]^[47] Alloying influences the shear modulus through mechanisms like solid solution strengthening, which causes slight increases by distorting the lattice and altering atomic interactions, typically on the order of a few percent for dilute additions.^[51] Precipitation hardening has a more pronounced effect, as the formation of second-phase particles with mismatched shear moduli creates local elastic incompatibilities that enhance overall rigidity. In aluminum-copper alloys, for instance, θ-phase precipitates can raise the effective modulus beyond what solid solution alone achieves.^[52] In single-crystal metals, the shear modulus displays strong anisotropy due to directional bonding preferences in the lattice. For copper, the shear modulus in the <100> direction is approximately 75 GPa, while it is about 23 GPa in the <111> direction (for the {111}<110> shear system), reflecting the softer response along close-packed planes.^[21]^[53] Temperature significantly affects the shear modulus in metals, with a general decrease as thermal expansion softens interatomic bonds. In the titanium alloy Ti-6Al-4V, the shear modulus drops from 44 GPa at 20°C to around 30 GPa at 600°C, highlighting its utility in high-temperature applications despite reduced stiffness.^[54] Comprehensive empirical data on these properties, including alloy-specific trends and phase effects, are compiled in the ASM Handbook Volume 2: Properties and Selection: Nonferrous Alloys and Special-Purpose Materials.^[49]

In Rocks and Composites

In rocks, the shear modulus varies significantly with lithology and microstructure, reflecting their heterogeneous and often porous nature. For instance, granite typically exhibits a shear modulus in the range of 20–30 GPa, as measured in laboratory tests on intact samples under moderate confining pressures, while sandstone shows lower values of approximately 5–15 GPa, influenced by its higher porosity and grain packing.^[55]^[56] Porosity plays a key role in reducing the effective shear modulus of the rock frame, as higher void content diminishes intergranular contacts and load-bearing capacity; this effect is incorporated into predictive models like the Gassmann relation, which, while primarily addressing bulk modulus changes due to fluid saturation, highlights how porosity degrades the dry-frame shear stiffness in fluid-bearing rocks.^[57] Confining pressure further modulates the shear modulus in rocks through mechanisms such as crack closure, which stiffens the material by reducing compliant pore space and microcracks. In sedimentary rocks, the shear modulus increases nonlinearly with pressure, with typical pressure derivatives (dG/dP) ranging from 0.5 to 1 for unconsolidated to semi-consolidated sediments, indicating substantial hardening at depths exceeding a few hundred meters.^[58] This pressure sensitivity is particularly pronounced in fractured or porous lithologies, where initial low-pressure compliance gives way to more isotropic behavior at higher effective stresses. Composite materials, such as fiber-reinforced polymers used in engineering applications, display shear moduli that depend on fiber orientation and matrix properties. In carbon-epoxy composites, the transverse shear modulus is around 5 GPa, reflecting the weaker matrix-dominated response perpendicular to the fibers, in contrast to higher longitudinal values. These properties are often approximated using the rule of mixtures, which weights the contributions of fiber and matrix shear moduli by volume fraction to estimate overall stiffness, providing a simple yet effective bound for design purposes.^[59]^[60] Anisotropy in rocks like shale arises from aligned bedding planes, leading to directional variations in shear modulus up to 50%, with minimum values parallel to bedding due to weak interlayer slip and maximum values perpendicular to it from enhanced resistance across layers. Specific examples illustrate these trends: basalt in the oceanic crust has a shear modulus of about 25 GPa, supporting its role in seismic wave propagation through the lithosphere, whereas fault gouge zones exhibit much lower values below 1 GPa (typically 0.1–0.3 GPa), signifying mechanical weakness and localization of deformation.^[61]^[62]^[63] Environmental factors, particularly fluid content, alter effective shear modulus in rocks; water saturation can lower it by 10–20% through reduced friction at grain contacts and increased pore pressure, exacerbating compliance in otherwise stiff formations. This saturation effect is leveraged in geophysical contexts, such as seismic mapping, to infer subsurface fluid distributions and rock integrity.^[64]

Advanced Models

Temperature-Dependent Models

The Varshni model provides an empirical description of the temperature dependence of the shear modulus in metals, given by

G(T) = G_0 - a \left( \frac{T}{\Theta} \right)^b \exp\left( -\frac{T}{\Theta} \right) + c T,

where G_0 is the shear modulus at 0 K, \Theta is the Debye temperature, and a, b, and c are fitting parameters. This form captures the softening at low temperatures through an Einstein-like anharmonic term while including a linear high-temperature contribution. It has been widely applied to fit experimental data for elastic constants in various metals, effectively modeling phonon contributions to modulus variation. The Chen-Gray model, developed within the Mechanical Threshold Stress (MTS) framework, employs a polynomial approximation for high-temperature behavior:

G(T) = G_0 (1 - \beta T^m),

where \beta and m are material-specific parameters tuned to experimental softening trends. This simple form emphasizes thermal expansion and anharmonic effects dominating at elevated temperatures, making it suitable for predictive simulations in high-strain-rate applications like shock loading in metals. It is often calibrated using room-temperature data and extrapolated for engineering steels. The Steinberg–Cochran–Guinan (SCG) model accounts for both pressure and temperature dependence, particularly useful for metals under extreme conditions. For temperature effects at constant pressure, it incorporates exponential softening and a term modeling the approach to zero modulus at the melting temperature T_m, often expressed in a form such as

G(T) = G_0 \exp\left[ -\lambda \frac{T - T_0}{T_0} \right] \left( \frac{T_m}{T_{m0}} \right)^n,

where \lambda, n, T_0, and T_{m0} are parameters, with T_m potentially adjusted for pressure. This enables modeling of modulus reductions near phase transitions or melting, improving accuracy in hydrodynamic simulations of material response under shock or high-temperature conditions compared to simpler polynomial models. Comparisons among these models reveal distinct suitability: the Varshni model excels for nuanced low-temperature fits in various metals, the Chen-Gray model for high-temperature softening in steels, and the SCG model for extreme pressure-temperature regimes, as demonstrated in fits to materials like AISI 4340 steel. Validation against density functional theory (DFT) simulations has shown reasonable agreement for these empirical models with ab initio predictions at elevated temperatures, enhancing their use in untested regimes. In recent years, machine learning approaches trained on DFT and experimental data have been developed to predict temperature-dependent shear moduli more efficiently, offering insights into novel materials.^[65]

Viscoelastic Relaxation Models

Viscoelastic materials exhibit time-dependent shear responses due to their combined elastic and viscous characteristics, particularly in polymers, where molecular rearrangements lead to relaxation processes following an applied shear stress. The shear relaxation modulus G(t) quantifies this behavior as the time-decaying stress response to a step strain, typically modeled using the generalized Maxwell model, which consists of multiple Maxwell elements in parallel with an equilibrium spring.^[66] In this framework, the relaxation modulus is expressed as

G(t) = G_\infty + \sum_{i=1}^N G_i \exp\left(-\frac{t}{\tau_i}\right),

where G_\infty is the long-term equilibrium modulus, G_i are the moduli of individual elements, and \tau_i are the relaxation times.^[67] This model captures the spectrum of relaxation times arising from diverse molecular motions, enabling prediction of stress decay over time scales from milliseconds to hours. In the frequency domain, oscillatory shear testing reveals the dynamic shear modulus G^*(\omega) = G'(\omega) + i G''(\omega), where G' is the storage modulus representing elastic energy storage, and G'' is the loss modulus indicating viscous energy dissipation.^[68] The loss tangent \tan \delta = G'' / G' serves as a measure of damping, with higher values signifying greater energy loss relative to storage.^[68] These components highlight the material's transition from glassy (high G', low \tan \delta) to rubbery states as frequency decreases or temperature rises. To fit experimental relaxation data, the Prony series representation is widely employed, discretizing the continuous relaxation spectrum into a sum of exponential terms with specific \tau_i and corresponding weights g_i = G_i / G_0, where G_0 is the instantaneous modulus.^[69] This approach facilitates numerical simulations in finite element analysis and accurate curve-fitting to time-domain measurements, ensuring the model aligns with observed decay profiles across multiple decades of time.^[70] For amorphous polymers, temperature influences relaxation times through the Williams-Landel-Ferry (WLF) equation, which provides a shift factor a_T to construct master curves via time-temperature superposition. The equation is given by

\log a_T = -\frac{C_1 (T - T_\mathrm{ref})}{C_2 + T - T_\mathrm{ref}},

where C_1 and C_2 are empirical constants (typically C_1 \approx 17.44, C_2 \approx 51.6 K near the glass transition temperature T_g), and T_\mathrm{ref} is a reference temperature.^[71] This superposition extends short-term data to long-term predictions, revealing how thermal activation accelerates segmental motions and broadens the relaxation spectrum. Representative examples illustrate these models in practice. In rubbers, such as natural rubber, the storage modulus G' is approximately 1 MPa at 1 Hz and room temperature, relaxing to about 0.1 MPa within seconds due to rapid disentanglement of polymer chains.^[72] Asphalt binders exhibit pronounced viscoelasticity, with shear relaxation moduli decreasing from initial values around 10^6 Pa to equilibrium levels below 10^4 Pa over minutes, influenced by oxidative aging and temperature.^[73] Biological tissues, like human orbital fat, show long-term relaxation moduli averaging 646 Pa at 100 s, reflecting fluid-like dissipation in soft matrices.^[74] At the molecular level, viscoelastic relaxation spectra stem from chain entanglements and glass transitions, where temporary topological constraints in polymer melts hinder reptation, leading to broad distributions of relaxation times.^[75] Below T_g, frozen segmental motions contribute to secondary relaxations, while above T_g, cooperative chain dynamics enable faster stress relief, as evidenced in entangled networks.^[76] These mechanisms underpin the time-dependent shear response, linking microstructure to macroscopic damping and recovery.^[77]

References

[1]
06mae324 - MP17
· For a material loaded in pure shear, the linear elastic relationship is: τ = G γ where G is the Shear modulus, and the shear strain, γ = tan θ = {w/L}
[2]
[PDF] 3. Shear Stress and Shear Strain
d) Material properties in shear – shear modulus. The relationship between shear stress and shear strain, as with those between normal stress and axial ...
[3]
[PDF] Materials and Elasticity Lecture M17: Engineering Elastic Constants
Consistent with the definition of the Young's modulus, the Shear modulus. G, is defined as: t = Gg. Again, note, that this relationship only holds if a pure ...
[4]
Strength and Stiffness Characteristics
Shear Rigidity: Also known as Modulus of Rigidity, is a measure of material resistance against shear distorsion. Its value is equal to the slope of the ...
[5]
unique relationship between small strain shear modulus and ...
Small strain shear modulus (G0) is an important dynamic soil property used in different aspect of geotechnical and earthquake engineering such as seismic site ...
[6]
[PDF] CHAPTER 2 - Stress and Strain
The off-diagonal elements of e cause shear strain; for example, in two-dimensions, if = 0 and there is no volume change, then au/ax = au./əz = 0, du,/əz дих ...
[7]
Thomas Young and the Theory of Structures 1807-2007
Young's Modulus. In 1807 Young introduced the world to the concept of an elastic modulus which defines the stiffness of a material:
[8]
Engineering Elasticity
First to measure the shear modulus in the modern sense. o ... Linear plots of stress versus strain begin to be widely used. o 1848 : Guillaume Wertheim ...
[9]
Relation between Young's modulus and shear modulus - DoITPoMS
where E is Young's modulus and ν is Poisson's ratio. Equating the diagonal strain in two cases gives the relation between Young's modulus E and shear modulus G:.
[10]
What is Poisson's ratio?
Poisson's ratio is related to elastic moduli K (also called B), the bulk modulus; G as the shear modulus; and E, Young's modulus, by the following (for ...
[11]
Lamé Constants -- from Eric Weisstein's World of Physics
where E is Young's modulus, is the Poisson ratio, G is the shear modulus, K is the bulk modulus, is the density, is P-wave speed, and is the S-wave speed.
[12]
Poisson's ratio range in linear isotropic classical elasticity
The mode structure allows one to distinguish isotropic from anisotropic ... Torsion modes are sensitive to the shear modulus, bending modes to Young's modulus ...
[13]
Introduction to atomman: ElasticConstants class
The ElasticConstants class represents a crystal's elastic constants, allowing setting/retrieval in various formats and handling transformations.
[14]
Mathematical foundation of elastic wave propagation - SEG Wiki
Sep 12, 2020 · The equation of wave propagation in elastic solids are derived by using Hooke's law and Newton's second law of motion.
[15]
SH Wave - an overview | ScienceDirect Topics
The polarization of the S-waves with respect to the free surface of the half-space is either vertical (for the SV-wave) or horizontal (for the SH wave). Only ...
[16]
Seismic Waves and Earth's Interior
An important distinguishing characteristic of an S-wave is its inability to propagate through a fluid or a gas because a fluids and gasses cannot transmit a ...
[17]
9.1: Understanding Earth Through Seismology
Apr 24, 2024 · S waves do not travel through liquids—they are stopped at the CMB—and there is an S wave shadow on the side of Earth opposite a seismic source.
[18]
Wave attenuation and internal friction as functions of frequency in ...
Mar 2, 2017 · Published and unpublished data on attenuation and internal friction in rocks are calculated and plotted against frequency in the range 10 ...
[19]
Elastic Moduli: a Tool for Understanding Chemical Bonding and ...
Dec 14, 2022 · In this review, we introduce the basics of elasticity and explain the origin of high-temperature lattice softening from a bonding perspective.
[20]
Temperature dependence of elastic and plastic deformation ...
Sep 9, 2020 · The above result corroborates the general trend of decreased stiffness in metals with increasing temperature.
[21]
Continuous model for the shear modulus as a function of pressure ...
Mar 1, 2003 · Continuous model for the shear modulus as a function of pressure and temperature up to the melting point: Analysis and ultrasonic validation.Missing: metals | Show results with:metals
[22]
[PDF] Elastic Properties of Metals and Alloys. II. Copper
Oct 29, 2009 · The earliest elastic study on cop- per was repQrted by Wertheim (1848). The first tryogenic. (S7 K) study was by Benton (1903) ~ The first ...
[23]
(PDF) Temperature dependence of the elastic constants of aluminum
Aug 7, 2025 · Over the temperature range investigated the isothermal bulk modulus and the two shear moduli have a simple exponential dependence on isobaric ...
[24]
Viscoelastic Materials - Rod Lakes
Viscoelastic materials are those for which the relationship between stress and strain depends on time or, in the frequency domain, on frequency.
[25]
[PDF] Optical Measurements of Frequency-Dependent Linear Viscoelastic ...
Feb 13, 1995 · This method probes the moduli over a greatly extended frequency range and provides significant new insight into the elastic susceptibility of ...
[26]
Anelasticity - an overview | ScienceDirect Topics
Anelastic dispersion means that the elastic constants of an Earth model found from seismology must be adjusted slightly to be appropriate for tidal frequencies.
[27]
[PDF] Physical Origins of Anelasticity and Attenuation in Rock
The resulting viscoelastic behavior is manifest in the dissipation of strain energy and associated frequency dependence of the shear modulus and wavespeed.
[28]
[PDF] Structural Metals Lab 1.2 53:086 - User pages
Torsion Testing of Structural Metals. Standards. • ASTM E143: Shear Modulus at Room Temperature. Purpose. • To determine the shear modulus of structural metals.
[29]
[PDF] Ultrasonic methods for determination of mechanical properties of ...
ultrasonic methods. In ourlaboratory, the pulse-echo method has been employed to measure the elastic characteristics of experimental composite quartz or ...
[30]
[PDF] Ultrasonic methods in determining elastic material properties of ...
Abstract—The presented study concerns the application of the pulse-echo ultrasound measurement technique in determining the elastic properties of fibres ...
[31]
Resonant ultrasound spectroscopy: The essential toolbox
Dec 17, 2019 · Resonant Ultrasound Spectroscopy (RUS) is an ultrasound-based minimal-effort high-accuracy elastic modulus measurement technique.
[32]
[PDF] Resonant Ultrasound Spectroscopy for Irregularly Shaped Samples ...
RUS measures the mechanical resonance spectrum of a three-dimensional solid. The resonance frequencies are determined by both intrinsic sample properties—the ...
[33]
[PDF] Dynamic Mechanical Analysis Basic Theory & Applications Training ...
The Modulus: Measure of materials overall resistance to deformation. Tan Delta: Measure of material damping. Increasing tan δ implies a greater potential for ...
[34]
Dynamic Mechanical Analysis ASTM D4065, D4440, D5279 - Intertek
Dynamic Mechanical Analysis (DMA) determines elastic modulus (or storage modulus, G'), viscous modulus (or loss modulus, G'') and damping coefficient (Tan D)
[35]
A Test Method for Experimental Determination of Shear Moduli of ...
One of the major problems in determining the through thickness shear properties of a composite material is the influence of stresses other than shear at the ...
[36]
Beam Torsion | Engineering Library
The maximum shear stress occurs at the outside surfaces of the beam and may be computed by setting r equal to ro in Equation (1-47). The maximum tensile and ...
[37]
Torsional Rigidity: Principles, Calculations, and Applications
Sep 9, 2024 · Torsional rigidity is denoted by the symbol GJ, where G stands for the shear modulus of the material, and J refers to the polar moment of ...
[38]
[PDF] Shear and Torsion Design of Prestressed .i.
The way in which these models can be used to design prestressed and non-prestressed concrete beams for torsion and shear is ex- plained. In addition, design ...
[39]
A nonlinear inversion method for Young's modulus and shear ...
Oct 22, 2024 · In geophysical inverse problems, the physical properties of the subsurface medium are inferred from observed data, which are often ill-posed ...Missing: rigidity | Show results with:rigidity
[40]
Effect of a compliant fault zone on the inferred earthquake slip ...
Jun 14, 2008 · ... fault embedded in a subspace of uniform shear modulus) is ϕ = equation image /μ* where μ* is the shear modulus μ on the fault. This choice ...
[41]
Seismic-scale finite element stress modeling of the subsurface
From left to right, the normal and shear stiffness of the central fault is decreasing over two orders of magnitude causing an increasing impact on the stress ...
[42]
[PDF] mechanical modeling of a fault-fold system, with application to the ...
The 1989 Loma Prieta earthquake resulted in several features that resemble the deformational features of earth- quakes associated with deep-seated reverse ...
[43]
Physical Models of Tissue in Shear Fields - ScienceDirect.com
The shear moduli of most soft tissues fall in the range 1–10 kPa. Serendipitously, that is the region in which the general wave equation becomes strongly ...
[44]
A Computational Model for Biomechanical Effects of Arterial ...
Mar 16, 2015 · A computational model was employed to investigate the effects of arterial compliance mismatch on blood flow, wall stress, and deformation.
[45]
[PDF] 5. Mohr-Coulomb Material Model - Rocscience
Using the Shear Strength Reduction (SSR) method this model can evaluate safety factors equivalent to those calculated based on limit equilibrium approach (Slide) ...
[46]
Mohr–Coulomb Failure Criterion | Rock Mechanics and Rock ...
Jul 3, 2012 · The Mohr–Coulomb (MC) failure criterion is a set of linear equations in principal stress space describing the conditions for which an isotropic material will ...
[47]
All About Shear Modulus | Xometry
Mar 23, 2023 · Shear modulus is the ratio of shear stress (force per unit area) to shear strain (displacement at the edge). It is an indication of a material's rigidity.Missing: physics | Show results with:physics
[48]
Modulus of Rigidity - The Engineering ToolBox
Shear Modulus (Modulus of Rigidity) is the elasticity coefficient for shearing or torsion force.Missing: 26 | Show results with:26
[49]
https://www.asminternational.org/asm-handbooks
[50]
https://link.aps.org/doi/10.1103/PhysRevB.85.054107
[51]
Elastic parameters of paramagnetic iron-based alloys from first ...
Feb 13, 2012 · The theoretical bulk modulus is 169.84 GPa for PM bcc Fe and 144.25 GPa for PM fcc Fe. The bulk modulus of PM bcc Fe is increased by V, Cr, and ...
[52]
Modeling the competition between solid solution and precipitate ...
In consideration of the composition-related lattice constant and shear modulus, the solid solution strengthening effect of various alloy elements is calculated ...
[53]
Precipitation hardening - Wikipedia
Modulus hardening results from the different shear modulus of the precipitate and the matrix, which leads to an energy change of dislocation line tension ...
[54]
Temperature Dependences of the Elastic Constants of Precipitation ...
For both alloys, the Young's moduli are about ten percent higher than for unalloyed aluminum, and they increase about ten percent on cooling from 300 to 4 K.Missing: metals | Show results with:metals<|separator|>
[55]
[PDF] MEASURING THE ELASTIC PROPERTIES OF ANISOTROPIC ...
Mar 19, 1994 · For copper and /Y-brass, the (11 I) indentation moduli are approximately. 10 and 25% larger than the { 100) modulus. The (110) moduli are ...
[56]
[PDF] MATERIALS DATA HANDBOOK H _ _" Titanium 6A1-4V +
Effect of temperature on the tensile and compressive modulus of solution-treated and a_ed alloy, figure 7.233. 7.14. Modulus of rigidity (shear modulus),. G. 7 ...
[57]
[PDF] The Effect of Temperature on the Static Elastic Moduli of Fractured ...
Feb 12, 2024 · Westerly granite, a commonly used standard plutonic igneous rock, has an intact bulk modulus of 39 GPa and a shear modulus of 28 GPa at 30 MPa ...
[58]
[PDF] ELASTIC PROPERTIES AND POROSITY
A critical porosity value exists which is typical for a given class of porous materials. ... Shear Modulus (GPa). Porosity. SANDSTONES. 0. 40. 80. 120. 0. 0.1.
[59]
Gassmann fluid substitutions: A tutorial - GeoScienceWorld
Jan 1, 2003 · Gassmann's equation relates the saturated bulk modulus of the rock to its porosity, the bulk modulus of the porous rock frame, the bulk modulus ...
[60]
(PDF) Pressure dependence of velocity and attenuation and its ...
Aug 10, 2025 · Closure of cracks with increasing confining pressure causes an increase in velocity and a decrease in attenuation. We present a form for the ...Missing: dG/ sediments
[61]
Carbon/Epoxy Composite Materials - Properties - AZoM
May 22, 2003 · Compressive Strength - Transverse, MPa, 570 ; Density, g.cm · 1.6 ; Shear modulus - in-plane, GPa, 5 ; Shear strength - in-plane, MPa, 90.
[62]
Rule-of-Mixture Equation - an overview | ScienceDirect Topics
The rule of mixtures equation refers to a set of equations used to predict the properties, such as modulus and strength, of fiber-reinforced composites ...
[63]
A study of the anisotropic static and dynamic elastic properties of ...
Oct 15, 2019 · The results suggest that the synthetic and natural shale indicate high similarity in the static and dynamic properties. The dynamic Young's ...Experimental Results · Dynamic Elastic Properties · Static Elastic Properties<|separator|>
[64]
Low‐Velocity Structure of Subducted Oceanic Crust in the Upper ...
Dec 19, 2023 · At ambient conditions, it has an adiabatic bulk modulus (KS0) of 46.9–69.4 GPa and shear modulus (G0) of 35.5–38.5 GPa, which are lower than ...
[65]
[PDF] In-situ rock tests for fault gouge zone：A case in Fengman ...
The deformation modulus of fault gouges ranged from 0.08 to 0.24 GPa with an average of 0.17 GPa, and the elastic modulus ranged from 0.18 to 0.38 GPa with an ...
[66]
[PDF] CHARACTERIZATION OF ROCK FOR CONSTRAINING ...
Water saturation decreases the shear modulus by 12 percent, again indicative of chemo- mechanical weakening. Q, is lower for the water saturated condition ...
[67]
[PDF] 10 Viscoelasticity
tE , the relaxation modulus for the Maxwell model, is η/. )( Et. Ee. tE ... (c) the generalized Maxwell model minus the free spring and free dash-pot.
[68]
Efficient and optimized identification of generalized Maxwell ...
The generalized Maxwell model had M = 3 Maxwell elements in parallel with a linear spring. The elastic modulus of the parallel spring was E0 = 100 MPa; the time ...
[69]
Viscoelasticity Imaging of Biological Tissues and Single Cells Using ...
In general, the storage modulus G′ reflects the shear elastic property while the loss modulus G″ reflects the viscous response of the material.
[70]
[PDF] Determining a Prony Series for a Viscoelastic Material From Time ...
In this study a method of determining the coefficients in a Prony series representation of a viscoelastic modulus from rate dependent data is presented.
[71]
[PDF] Prony series calculation for viscoelastic behavior modeling of ...
Viscoelastic properties of structural adhesives can be incorporated to CAE by es- timating Relaxation modulus in function of time through Prony series by ...
[72]
Application of a Universal and Developed WLF Equation - PMC - NIH
A developed WLF (DWLF) equation was established to describe the temperature dependence of relaxation property for the polymer ranging from secondary relaxation ...
[73]
[PDF] Introduction to Dynamic Mechanical Analysis and its Application to ...
The storage modulus G' and tan δ were measured at a frequency of 1 Hz and a strain of 0,07% at temperatures from -120 °C to 130 °C.
[74]
Effect of model compounds on stress relaxation in asphalt binders
Dec 27, 2024 · Wax accelerates stress relaxation in asphalt binders. Asphaltene increases relaxation modulus significantly at 4 % dosage and slow cooling.
[75]
Shear viscoelastic properties of human orbital fat - ScienceDirect.com
The long-term relaxation modulus averaged 646 ± 264 Pa at 100 s. Frequency sweep testing validated the parameters of the Prony series fitted to the experimental ...
[76]
Molecular Model for Linear Viscoelastic Properties of Entangled ...
Oct 15, 2024 · A molecular Kuhn-scale model is presented for the stress relaxation dynamics of entangled polymer networks.
[77]
Probing the Molecular Mechanism of Viscoelastic Relaxation in ...
Dec 1, 2023 · This review introduces our recent progress on the studies on the molecular mechanism of viscoelasticity in transient networks using multiple methods and model ...
[78]
Cooperative Intramolecular Dynamics Control the Chain-Length ...
May 27, 2022 · Our study provides a new understanding of molecular relaxations and the glass transition in polymers, which paves the way for predictive design ...<|control11|><|separator|>