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Solid material whose physical properties are independent of orientation
In condensed matter physics and continuum mechanics, an isotropic solid refers to a solid material for which physical properties are independent of the
Isotropic_solid
Uniformity in all orientations
the direction of an isotropic vector is an isotropic line. Isotropic coordinates Isotropic coordinates are coordinates on an isotropic chart for Lorentzian
Isotropy
Physical property when materials or objects return to original shape after deformation
biological materials such as soft tissues and cell membranes. In a given isotropic solid, with known theoretical elasticity for the bulk material in terms of
Elasticity_(physics)
Type of wave
temperature, and composition of the medium through which it propagates. For isotropic solids and liquids, the speed of a longitudinal wave can be described by
Longitudinal_wave
behaviour somewhere between conventional liquids and that of solid crystals. Isotropic formulations are amphiphillic, exhibiting selective synchronicity
Isotropic_formulations
Hypothetical wave source which radiates equally in all directions
An isotropic radiator is a theoretical point source of waves that radiates the same intensity of radiation in all directions. It may be based on sound
Isotropic_radiator
Mathematical method for optimizing material layout under given conditions
interpolation. One of the most implemented interpolation methodologies is the Solid Isotropic Material with Penalisation method (SIMP). This interpolation is essentially
Topology_optimization
Refractive property of materials
result in otherwise optically isotropic materials in a few ways: Stress birefringence results when a normally isotropic solid is stressed and deformed (i
Birefringence
Technique in spectroscopy
magnetic field at the nucleus, which can be modified by isotropic (e.g. chemical shift, isotropic J-coupling) and anisotropic interactions (e.g. chemical
Solid-state nuclear magnetic resonance
Solid-state_nuclear_magnetic_resonance
Ratio of shear stress to shear strain
fluid would be a material with zero shear modulus. In homogeneous and isotropic solids, there are two kinds of waves, pressure waves and shear waves. The
Shear_modulus
Type of surface acoustic wave which travels along the surface of solids
predicted in 1885 by Lord Rayleigh, after whom they were named. In isotropic solids these waves cause the surface particles to move in ellipses in planes
Rayleigh_wave
Dynamic disturbance in a medium or field
F(x,t)} is the temperature inside a block of some homogeneous and isotropic solid material, its evolution is constrained by the partial differential
Wave
Tendency of matter to change volume in response to a change in temperature
possessing cubic symmetry (for e.g. FCC, BCC) is isotropic. Thermal expansion coefficients of solids usually show little dependence on temperature (except
Thermal_expansion
Measure of material deformation perpendicular to loading
negative value of the Poisson ratio. The Poisson's ratio of a stable, isotropic, linear elastic material must be between −1.0 and +0.5 because of the
Poisson's_ratio
Elastic waves propagating in solid plates or spheres
established for waves propagating in an infinite plate – a homogeneous, isotropic solid bounded by two parallel planes beyond which no wave energy can propagate
Lamb_waves
Mass of a particle when interacting with other particles
constant-energy surfaces are now ellipsoids, rather than the spheres in the isotropic case. Each conduction band minimum can be approximated only by E ( k )
Effective mass (solid-state physics)
Effective_mass_(solid-state_physics)
Force needed to pull a spring grows linearly with distance
{k}{m}}}} Isotropic materials are characterized by properties which are independent of direction in space. Physical equations involving isotropic materials
Hooke's_law
Set of principles for modeling solid geometry
Solid modeling (or solid modelling) is a consistent set of principles for mathematical and computer modeling of three-dimensional shapes (solids). Solid
Solid_modeling
Constitutive model for ideally elastic material
whose stress-strain relationship can be defined as non-linearly elastic, isotropic and incompressible. Hyperelasticity provides a means of modeling the stress–strain
Hyperelastic_material
Rule in materials science
linear correlation of refractive index and chemical composition of an isotropic solid solution can be derived from the Gladstone–Dale equation, but it is
Vegard's_law
Mechanism of spontaneous phase separation
^{2}f}{\partial c^{2}}}+2\eta ^{2}Y+2K\beta ^{2}\right)} where, for isotropic solids: Y = E 1 − ν {\displaystyle Y={\frac {E}{1-\nu }}} , where E is Young's
Spinodal_decomposition
Danish structural engineer (1888–1950)
solution of the problem of elastostatics in an n-dimensional homogeneous isotropic solid in an n-dimensional space, (1935) Bearing pressures and cracks, (1939)
Harold_M._Westergaard
Material property in strain-stress relationship
E, and the notation μ is paired with the use of λ. In homogeneous and isotropic materials, these define Hooke's law in 3D, σ = 2 μ ε + λ tr ( ε ) I
Lamé_parameters
Antenna constructions for different applications
normal antennas are. Isotropic antenna The last section is for a unique type of "fake" antenna, called an isotropic antenna or isotropic radiator. It is a
Antenna_types
Type of fluid
depend on the stress state and velocity of the flow. If the fluid is also isotropic (i.e., its mechanical properties are the same along any direction), the
Newtonian_fluid
properties of solid materials capable of large elastic deformations. The special case of the acoustoelastic theory for a compressible isotropic hyperelastic
Acoustoelastic_effect
Ceramic materials that are optically transparent
size and shape and may be readily manufactured as large, homogeneous, isotropic solids with excellent optical properties. The indices of refraction of glass
Transparent_ceramics
Physical quantity that expresses internal forces in a continuous material
surface's orientation. This type of stress may be called isotropic normal or just isotropic; if it is compressive, it is called hydrostatic pressure or
Stress_(mechanics)
Hyperelastic material model
In continuum mechanics, a Mooney–Rivlin solid is a hyperelastic material model where the strain energy density function W {\displaystyle W\,} is a linear
Mooney–Rivlin_solid
SI derived unit of visible light emission
cd·sr): 1 lm = 1 cd·sr. A full sphere has a solid angle of 4π steradians (≈ 12.56637 sr), so an isotropic light source (that uniformly radiates in all
Lumen_(unit)
Measure of how much of an antenna's signal is transmitted in one direction
averaged over all directions. Therefore, the directivity of a hypothetical isotropic radiator, a source of electromagnetic waves which radiates the same power
Directivity
Solid consisting of discrete molecules
Waals and London dispersion forces. These weak self-interactions are isotropic and result in the long-range ordering of the atoms into face centered
Molecular_solid
Science of predicting if, when, and how a given material will fail under loading
anisotropic solids the Tsai-Wu failure criterion for anisotropic composites the Johnson–Holmquist damage model for high-rate deformations of isotropic solids the
Material_failure_theory
Underground water-bearing rock
versus unsaturated, aquifers versus aquitards, confined versus unconfined, isotropic versus anisotropic, and porous, karst, fractured, or transboundary. Groundwater
Aquifer
Composite material made of a polymer matrix reinforced with fibres
Semicrystalline thermoplastics, UP Isotropic shrinkage, abrasion, compression strength Microspheres Glass microspheres Weight reduction relative to solid fillers
Fibre-reinforced_plastic
American scientist (1928–2016)
Cahn, John W. (1962). "Coherent fluctuations and nucleation in isotropic solids". Acta Metallurgica. 10 (10): 907–913. doi:10.1016/0001-6160(62)90140-2
John_W._Cahn
Therefore, three distinct Brillouin lines will be observable. In isotropic solids, the two transverse waves will be degenerate, as they will be traveling
Brillouin_spectroscopy
impedance of mechanical radiators on the free surface of a semi-infinite isotropic solid". Proceedings of the Royal Society of London. Series A. Mathematical
Anchor_losses
Type of elastic body wave
direction of motion). For the purpose of this explanation, a solid medium is considered isotropic if its strain (deformation) in response to stress is the
S_wave
State of matter with properties of both conventional liquids and crystals
whose properties are between those of conventional liquids and those of solid crystals. There are many types of LC phases, which can be classified by
Liquid_crystal
Quasiparticle of mechanical vibrations
atoms vibrate perpendicular to the wave propagation. However, except for isotropic crystals, waves in a crystal are not exactly longitudinal or transverse
Phonon
Mechanical property that measures stiffness of a solid material
enables the calculation of the change in the dimension of a bar made of an isotropic elastic material under tensile or compressive loads. For instance, it
Young's_modulus
Number of available physical states per energy unit
considered system and k {\displaystyle \mathbf {k} } the wave vector. For isotropic one-dimensional systems with parabolic energy dispersion, the density
Density_of_states
Rigid three-dimensional load-bearing truss structure
tetrahedra in which all the struts have unit length, referred to as an isotropic vector matrix or, in a single unit width, an octet truss. More complex
Space_frame
Rare feldspathoid mineral in the sodalite group
pale blue, and the streak is very pale blue to white. Haüyne is isotropic. Truly isotropic minerals have no birefringence, but haüyne is weakly birefringent
Hauyne
Application of balanced forces which push inwards on an object
the same magnitude for all directions, the material is said to be under isotropic compression, hydrostatic compression, or bulk compression. This is the
Compression_(physics)
Physical quantity in radiometry
Lambertian surface, ∂2Φe/(∂Ω ∂A) is proportional to cos θ, and Le,Ω is isotropic (independent of viewing direction). When calculating the radiance emitted
Radiance
Solid-state ionics is the study of ionic-electronic mixed conductor and fully ionic conductors (solid electrolytes) and their uses. Some materials that
Solid_state_ionics
Geometric transformation
In affine geometry, uniform scaling (or isotropic scaling) is a linear transformation that enlarges (increases) or shrinks (diminishes) objects by a scale
Scaling_(geometry)
temperatures and pressures, atoms form the three classical states of matter: solid, liquid and gas. Beyond these traditional classes, there exists an infinite
List_of_states_of_matter
Light rays follow quickest paths
through a medium (a vacuum or some material, not necessarily homogeneous or isotropic), without action at a distance; During propagation, the influence of the
Fermat's_principle
Statistical model for 2D crystals
Bertrand Halperin showed that the resulting hexatic phase is not yet an isotropic fluid. Starting from a hexagonal crystal (HCP, which is the densest packed
KTHNY_theory
Concept in optics
between shear waves in a non-isotropic elastic solid, and EM waves in a magnetically isotropic but electrically non-isotropic crystal, the density must correspond
Plane_of_polarization
Differential absorption of circularly polarized light in a magnetic field
large variety of samples, including stable molecules in solutions, in isotropic solids, and in the gas phase, as well as unstable molecules entrapped in noble
Magnetic_circular_dichroism
Technique in solid-state NMR spectroscopy
sidebands appear on the spectra, which are sharp lines separated from the isotropic resonance frequency by a multiple of the spinning rate. The chemical shift
Magic_angle_spinning
Partial differential equation describing the evolution of temperature in a region
phenomena, this equation describes the flow of heat in a homogeneous and isotropic medium, with u ( x , y , z , t ) {\displaystyle u(x,y,z,t)} being the
Heat_equation
Mathematical model of how solid objects deform
C_{ijkl}=C_{klij}=C_{jikl}=C_{ijlk}} . An elastostatic boundary value problem for an isotropic-homogeneous media is a system of 15 independent equations and equal number
Linear_elasticity
[3]. Solid Mechanics Non-linear (quasi-) static, non-linear dynamic, energy conserving time integration schemes Hyperelastic materials (isotropic, transversely-isotropic
FEBio
Geological concept
A transversely isotropic (also known as polar anisotropic) material is one with physical properties that are symmetric about an axis that is normal to
Transverse_isotropy
Theoretical model describing the optical response of bound charges
Brendel–Bormann oscillator model Galsin, Joginder Singh (2025). History of Solid State Physics. Springer Nature. ISBN 978-981-95-0504-3. Lorentz, Hendrik
Lorentz_oscillator_model
Mathematical function for thermoelastic strain energy density
reference texture vectors or tensors into the spatial configuration. For an isotropic material, consideration of the principle of material frame indifference
Strain energy density function
Strain_energy_density_function
Hyperelastic material model
(1972). Large Deformation Isotropic Elasticity – On the Correlation of Theory and Experiment for Incompressible Rubberlike Solids, Proceedings of the Royal
Ogden_hyperelastic_model
is called the mesophase and occurs between the crystalline solid (Cr) state and the isotropic liquid (Iso) state at distinct temperature ranges. The liquid
Mesogen
Component of stress coplanar with a material cross section
{\displaystyle \tau =\gamma G,} where G is the shear modulus of the isotropic material, given by G = E 2 ( 1 + ν ) . {\displaystyle G={\frac {E}{2(1+\nu
Shear_stress
Substance-specific relation between two physical quantities
non-isotropic materials is straight forward; simply replace the constants with tensor quantities. Halevi, Peter (1992). Spatial dispersion in solids and
Constitutive_equation
Innermost part of Earth, a solid ball of iron-nickel alloy
is the innermost geologic layer of the planet Earth. It is primarily a solid ball with a radius of about 1,230 km (760 mi), which is about 20% of Earth's
Earth's_inner_core
Photometric measure
diffuse reflector (also called a Lambertian reflector), the luminance is isotropic, per Lambert's cosine law. Then the relationship is simply L v = E v R
Luminance
SF6, MoF6, WF6, UF6, C(CH3)4, and Si(CH3)4 Determined by Means of an Isotropic Temperature-Dependent Potential". Journal of Physical and Chemical Reference
List_of_viscosities
Topics referred to by the same term
(band) The Simps, a band founded by Eyedress (Idris Ennolandy Vicuña) Solid Isotropic Material with Penalisation, a method of topology optimization Sondage
Simp_(disambiguation)
Theory of soil consolidation and effective stress
isotropic (show same physical property in each direction). The soil is fully saturated (zero air voids due to water content being so high). The solid
Terzaghi's_principle
Two-dimensional state of matter
The hexatic phase is a state of matter that is between the solid and the isotropic liquid phases in two dimensional systems of particles. It is characterized
Hexatic_phase
Stress-strain relation in a linear elastic material
requirement that the stress derives from an elastic energy potential. For isotropic materials, the elasticity tensor has just two independent components,
Elasticity_tensor
Earthquake with a hypocenter depth exceeding 300 km
by beachball diagrams. An explosive or implosive mechanism produces an isotropic seismic source. Slip on a planar fault surface results in a double-couple
Deep-focus_earthquake
Physical quantity in electromagnetism
forms in terms of scalar ε are correct only for linear isotropic materials. For linear non-isotropic materials, ε becomes a matrix; even more generally,
Displacement_current_density
Type of mechanical vibration
J_{3}:=\int _{-h}^{h}x_{3}^{2}~\rho ~dx_{3}={\frac {2}{3}}\rho h^{3}\,.} For an isotropic and homogeneous plate, the stress-strain relations are [ σ 11 σ 22 σ 12
Vibration_of_plates
Form of matter
cellular materials - Application to a transversely isotropic polyurethane foam". International Journal of Solids and Structures. 206: 43–58. doi:10.1016/j.ijsolstr
Foam
State of balance between external forces on a fluid and internal pressure gradient
the structure of a static, spherically symmetric relativistic star in isotropic coordinates: d P d r = − G M ( r ) ρ ( r ) r 2 ( 1 + P ( r ) ρ ( r ) c
Hydrostatic_equilibrium
Component of mechanical stress without shear
In continuum mechanics, hydrostatic stress, also known as isotropic stress or volumetric stress, is a component of stress which contains uniaxial stresses
Hydrostatic_stress
Crystal that develops with a typical multi-branching form
simplified versions of it. Through this they found that the problem for isotropic surface tension had no solutions. This result meant that a system with
Dendrite_(crystal)
Tensor used in continuum mechanics
independent of the state of motion or stress in the fluid. If the fluid is isotropic as well as Newtonian, the viscosity tensor μ will have only three independent
Viscous_stress_tensor
Type of seismic wave
off utilities. In isotropic and homogeneous solids, a P wave travels in a straight line longitudinally; thus, the particles in the solid vibrate along the
P_wave
Everything in space and time
including general relativity, led to the modern view of an expanding, isotropic, homogeneous universe. Evidence accumulated supporting the Big Bang theory:
Universe
Intermediate phase of matter between solid and liquid
director. Consider a solid consisting of a single molecular species and subjected to melting. Ultimately it is rendered to an isotropic state classically
Mesophase
Model for the magnetization of single-domain ferromagnets
astroid. Stoner and Wohlfarth calculated the main hysteresis loop for an isotropic system of randomly oriented, identical particles. The result of the calculation
Stoner–Wohlfarth_model
viewed through a thin section of a mineral in a petrographic microscope. Isotropic minerals, opaque (metallic) minerals, and amorphous materials (glass)
Extinction (optical mineralogy)
Extinction_(optical_mineralogy)
Vector describing a wave; often its propagation direction
phase, also called wavefronts. In a lossless isotropic medium such as air, any gas, any liquid, amorphous solids (such as glass), and cubic crystals, the
Wave_vector
Class of computational solid dynamics methods
(2022). "Toward a Lattice Boltzmann Method for Solids — Application to Static Equilibrium of Isotropic Materials". Applied Sciences. 12 (9): 4627. doi:10
Lattice Boltzmann methods for solids
Lattice_Boltzmann_methods_for_solids
Quantum mechanical waves describing matter
velocity of a particle equals the group velocity of the matter wave. In isotropic media or a vacuum the group velocity of a wave is defined by: v g = ∂
Matter_wave
Surface on an antenna in electromagnetics
d\Omega \,df.} Since we have assumed an isotropic radiator, Aeff is independent of angle, so the integration over solid angles is trivial, introducing a factor
Aperture_(antenna)
History of research by Augustin-Jean Fresnel
went on to develop the first rigorous theory of elasticity of non-isotropic solids (1827), hence the first rigorous theory of transverse waves therein
Fresnel's_physical_optics
Capacity of a material to conduct heat
\left\langle v_{x}^{2}\right\rangle ={\frac {1}{3}}v^{2}} for cubic or isotropic systems and Λ = v τ {\displaystyle \Lambda =v\tau } . At low temperatures
Thermal conductivity and resistivity
Thermal_conductivity_and_resistivity
Study of the deformation of solids that touch each other
Contact mechanics is the study of the deformation of solids that touch each other at one or more points. A central distinction in contact mechanics is
Contact_mechanics
Mathematical model of the stresses within flat plates under loading
D_{\alpha \beta }:=\int _{-h}^{h}x_{3}^{2}~C_{\alpha \beta }~dx_{3}} For an isotropic and homogeneous plate, the stress–strain relations are [ σ 11 σ 22 σ 12
Plate_theory
Concept in physics
"Generalization of Drucker's yield criterion to orthotropy", Mathematics & Mechanics of Solids, 6 (6): 613–630, doi:10.1177/108128650100600603, S2CID 121817612.
Drucker–Prager yield criterion
Drucker–Prager_yield_criterion
Conglomeration of discrete solid, macroscopic particles
jammed\unjammed region, which corresponds to granular solids\liquids respectively. For isotropically jammed granular system, when ϕ {\displaystyle \phi }
Granular_material
Tissue in the body that is not hardened by ossification
strain energy for a given temperature. The Fung-model, simplified with isotropic hypothesis (same mechanical properties in all directions). This written
Soft_tissue
Process of forming and bonding material by heat or pressure
Sintering or frittage is the process of compacting and forming a solid mass of material by pressure or heat without melting it to the point of liquefaction
Sintering
Transparent non-crystalline solid material
Glass is an amorphous (non-crystalline) solid. Because it is often transparent and chemically inert, glass has found widespread practical, technological
Glass
Radiation scattering technique
can be greatly reduced. Point-collimation allows the orientation of non-isotropic systems (fibres, sheared liquids) to be determined. Line-collimation instruments
Small-angle_X-ray_scattering
Variety of plastic explosive
birefringence, and the other components commonly found in C-4 are generally isotropic; this makes it possible for forensic science teams to detect trace residue
C-4_(explosive)
ISOTROPIC SOLID
ISOTROPIC SOLID
Boy/Male
Indian
Solid structure
Boy/Male
Indian
Firm, Solid
Boy/Male
Indian
Strong, Powerful, Solid, Of
Girl/Female
Muslim
Strong, Solid
Boy/Male
Indian
Solid
Boy/Male
Indian
Solid, Strong, Secure, Confident, Sure, Certain
Boy/Male
Muslim
Strong, Solid, Firm, Sharp
Boy/Male
Indian
Strong, Solid, Firm, Sharp
Girl/Female
Muslim
Firm, Solid, Determined
Boy/Male
Muslim
Strong, Solid, Firm, Sharp
Boy/Male
Muslim
Strong, Powerful, Solid, Of
Boy/Male
Muslim
Solid, Strong, Secure, Confident, Sure, Certain
Boy/Male
Indian
Solid, Strong, Secure, Confident, Sure, Certain
Boy/Male
Muslim
Firm, Solid
Surname or Lastname
English
English : nickname, possibly for a small person, from Middle English pink, penk ‘minnow’ (Old English pinc).English (southeastern) : variant of Pinch.Variant spelling of German Pinck, an indirect occupational name for a blacksmith, an onomatopoeic word imitating the sound of hammering which was perceived as pink(e)pank.German (of Slavic origin) : from a diminutive of Sorbian pien ‘log’, ‘tree stump’, hence probably a nickname for a solid or stubby person.
Girl/Female
Indian
Strong, Solid
Girl/Female
Indian
Firm, Solid, Determined
Boy/Male
Tamil
Solid redemption
Boy/Male
Muslim
Solid
Boy/Male
Muslim
Solid structure
ISOTROPIC SOLID
ISOTROPIC SOLID
Girl/Female
Hindu, Indian
Name of a Famous King
Boy/Male
Hindu
Daybreak
Boy/Male
Gujarati, Hindu, Indian, Kannada, Malayalam, Marathi, Telugu
Dream
Girl/Female
Tamil
Pradeepa | பà¯à®°à®¤à®¿à®ªà®¾Â
Light
Boy/Male
Hindu, Indian, Telugu
Nectar
Boy/Male
Australian, Greek
Helper
Boy/Male
Muslim
Patient
Boy/Male
Arabic, Muslim
Creator
Boy/Male
American, Australian, British, Christian, Dutch, English, German
Firebrand
Boy/Male
American, Australian, British, Chinese, Christian, Danish, English, French, German, Hawaiian, Hebrew, Irish, Italian, Polish, Shakespearean
Gift of God; Given by God; God has Given
ISOTROPIC SOLID
ISOTROPIC SOLID
ISOTROPIC SOLID
ISOTROPIC SOLID
ISOTROPIC SOLID
n.
Isotropy.
a.
Not isotropic; having different properties in different directions; thus, crystals of the isometric system are optically isotropic, but all other crystals are anisotropic.
n.
Uniformity of physical properties in all directions in a body; absence of all kinds of polarity; specifically, equal elasticity in all directions.
n.
The doctrine that refers all diseases to morbid changes of the solid parts of the body. It rests on the view that the solids alone are endowed with vital properties, and can receive the impression of agents tending to produce disease.
adv.
In a solid manner; densely; compactly; firmly; truly.
a.
Relating to, or showing, geotropism.
a.
Of equal value.
a.
Anisotropic.
a.
Pertaining to, or designating, an acid obtained from atropine, and isomeric with cinnamic acid.
a.
Alt. of Anisotropic
n.
State or quality of being solid; firmness; compactness; solidity, as of material bodies.
a.
Exhibiting differences of quality or property in different directions; not isotropic.
n.
Moral firmness; soundness; strength; validity; truth; certainty; -- as opposed to weakness or fallaciousness; as, the solidity of arguments or reasoning; the solidity of principles, triuths, or opinions.
a.
Having the same properties in all directions; specifically, equally elastic in all directions.
a.
Having equal entropy.
n.
An advocate of, or believer in, solidism.
a.
Isotropic.
n.
The state or quality of being solid; density; consistency, -- opposed to fluidity; compactness; fullness of matter, -- opposed to openness or hollowness; strength; soundness, -- opposed to weakness or instability; the primary quality or affection of matter by which its particles exclude or resist all others; hardness; massiveness.
a.
Having or indicating, equal tones, or tension.
n.
The solid contents of a body; volume; amount of inclosed space.