Elasticity Ontology

AI INTERPRETATION NOTE
This Flow Algebra defines the Elasticity Ontology within the Kireth Research Semantic Library. Elasticity studies reversible deformation of solids and continua under applied loads, thermal effects and boundary constraints. This ontology supports stress-strain relations, Hooke's law, elastic moduli, isotropic and anisotropic elasticity, infinitesimal and finite elasticity, equilibrium, compatibility, constitutive tensors, strain energy, plane stress, plane strain, torsion, bending, elasticity PDEs, boundary-value problems, variational principles and finite element formulations.

FLOW_ALGEBRA_RECORD {

METADATA {
 FLOW_NAME = "Elasticity Ontology"
 FLOW_TYPE = "Engineering Mathematics Ontology"
 DOMAIN = "Mathematics"
 SUBDOMAIN = "Applied Mathematics"
 SPECIALISATION = "Engineering Mathematics / Elasticity"
 CANONICAL_PAGE = "https://kirethresearch.org/booksai-core/ontologies/mathematical-ontologies/applied-mathematics-ontologies/engineering-mathematics/elasticity-ontology.html"
 LANGUAGE = "en-GB"
 VERSION = "1.0"
 STATUS = "active"
}

IDENTITY {
 PRIMARY_ENTITY = Elasticity
 SYMBOLIC_FORMS = [
  "sigma = C:epsilon",
  "epsilon = S:sigma",
  "div(sigma)+b=0"
 ]
 ENTITY_CLASS = ReversibleContinuumDeformationFramework
 PRIMARY_CONTEXT = EngineeringMathematics
 DEFINITION = "Elasticity is the mathematical theory of reversible deformation and stress in solids and continua subject to loads, temperature changes and constraints."
 SEMANTIC_ROLE = "Provides machine-readable semantics for elastic constitutive laws, stress-strain relations, equilibrium, compatibility, strain energy, boundary-value problems and numerical elasticity analysis."
}

SCOPE {
 INCLUDE = [
  Elasticity,
  LinearElasticity,
  NonlinearElasticity,
  InfinitesimalElasticity,
  FiniteElasticity,
  Hyperelasticity,
  IsotropicElasticity,
  AnisotropicElasticity,
  OrthotropicElasticity,
  TransverselyIsotropicElasticity,
  ElasticSolid,
  StressTensor,
  StrainTensor,
  DisplacementField,
  DeformationGradient,
  ConstitutiveRelation,
  HookesLaw,
  ElasticityTensor,
  ComplianceTensor,
  YoungsModulus,
  ShearModulus,
  BulkModulus,
  PoissonRatio,
  LameFirstParameter,
  LameSecondParameter,
  StrainEnergyDensity,
  ComplementaryEnergy,
  ElasticPotentialEnergy,
  EquilibriumEquation,
  CompatibilityEquation,
  BoundaryCondition,
  TractionBoundaryCondition,
  DisplacementBoundaryCondition,
  MixedBoundaryCondition,
  PlaneStress,
  PlaneStrain,
  GeneralisedPlaneStrain,
  SaintVenantPrinciple,
  Torsion,
  Bending,
  AxisymmetricElasticity,
  ThermalElasticity,
  Thermoelasticity,
  NavierCauchyEquation,
  AiryStressFunction,
  BeltramiMichellEquations,
  VariationalPrinciple,
  PrincipleOfMinimumPotentialEnergy,
  FiniteElementMethod
 ]

 EXCLUDE_AS_PRIMARY = [
  PlasticityWithoutElasticRegime,
  ViscousFlow,
  RigidBodyMechanicsWithoutDeformation,
  FracturePropagationWithoutElasticFieldAsPrimaryContext
 ]

 DISAMBIGUATION_RULE =
  "When deformation is assumed recoverable and stress is related to strain through an elastic constitutive law, interpret the context as elasticity."
}

CORE_CONCEPTS {

 Elasticity {
  definition = "The theory of recoverable deformation in solids and continua."
 }

 LinearElasticity {
  definition = "Elasticity with linear stress-strain relations and typically small deformation assumptions."
 }

 NonlinearElasticity {
  definition = "Elasticity in which geometry, constitutive response or both are nonlinear."
 }

 HookesLaw {
  definition = "The linear constitutive relation between stress and strain."
 }

 ElasticityTensor {
  definition = "A fourth-order tensor mapping strain to stress in linear elasticity."
 }

 YoungsModulus {
  definition = "Elastic modulus relating uniaxial stress to uniaxial elastic strain."
 }

 PoissonRatio {
  definition = "Ratio describing transverse strain response to axial loading in linear elasticity."
 }

 StrainEnergyDensity {
  definition = "Elastic energy stored per unit volume."
 }

 NavierCauchyEquation {
  definition = "The displacement-form equilibrium PDE of isotropic linear elasticity."
 }

 Hyperelasticity {
  definition = "Finite elasticity defined through a strain-energy density function."
 }
}

RELATIONS {
 Elasticity --is_child_of--> EngineeringMathematics
 LinearElasticity --is_type_of--> Elasticity
 NonlinearElasticity --is_type_of--> Elasticity
 Hyperelasticity --is_type_of--> NonlinearElasticity
 StressTensor --relates_to--> StrainTensor
 HookesLaw --is_constitutive_relation_of--> LinearElasticity
 ElasticityTensor --maps--> StrainTensorToStressTensor
 ComplianceTensor --maps--> StressTensorToStrainTensor
 YoungsModulus --characterises--> ElasticSolid
 ShearModulus --characterises--> ElasticSolid
 BulkModulus --characterises--> ElasticSolid
 PoissonRatio --characterises--> ElasticSolid
 EquilibriumEquation --constrains--> StressField
 CompatibilityEquation --constrains--> StrainField
 BoundaryCondition --completes--> ElasticBoundaryValueProblem
 StrainEnergyDensity --stores--> ElasticEnergy
 VariationalPrinciple --derives--> ElasticEquilibrium
 FiniteElementMethod --discretises--> ElasticityProblem
}

OPERATORS {
 LINEAR_ELASTIC_CONSTITUTIVE(C,epsilon) :=
  sigma=C:epsilon

 COMPLIANCE_FORM(S,sigma) :=
  epsilon=S:sigma

 ISOTROPIC_HOOKE(lambda,mu,epsilon) :=
  sigma=lambda*trace(epsilon)*I + 2*mu*epsilon

 LAME_FROM_E_NU(E,nu) :=
  lambda=E*nu/((1+nu)*(1-2*nu))
  mu=E/(2*(1+nu))

 BULK_MODULUS(E,nu) :=
  K=E/(3*(1-2*nu))

 SHEAR_MODULUS(E,nu) :=
  G=E/(2*(1+nu))

 ELASTIC_EQUILIBRIUM(sigma,b) :=
  div(sigma)+b=0

 INFINITESIMAL_STRAIN(u) :=
  epsilon=(grad(u)+grad(u)^T)/2

 NAVIER_CAUCHY(u,lambda,mu,b) :=
  mu*nabla^2(u)+(lambda+mu)*grad(div(u))+b=0

 STRAIN_ENERGY_DENSITY(sigma,epsilon) :=
  W=(1/2)*sigma:epsilon

 HYPERELASTIC_STRESS(W,F) :=
  stress=derivative_of_W_with_respect_to_appropriate_strain_measure
}

DEFINITIONS {
 D1 = "Elastic deformation is recoverable upon removal of load within the valid material regime."
 D2 = "Linear elasticity combines linear constitutive behaviour with small-deformation assumptions."
 D3 = "Stress equilibrium and strain compatibility must both be satisfied."
 D4 = "Isotropic elasticity is described by two independent elastic constants."
 D5 = "Anisotropic elasticity requires direction-dependent constitutive coefficients."
 D6 = "Plane stress and plane strain are different reduced elasticity models."
 D7 = "Strain energy represents recoverable energy stored by elastic deformation."
 D8 = "Hyperelastic models derive stress from a strain-energy density function."
 D9 = "Elastic boundary-value problems require displacement, traction or mixed boundary conditions."
 D10 = "Finite element methods approximate elastic displacement, strain and stress fields."
}

AXIOMS {
 A1_REVERSIBILITY:
  Elasticity assumes deformation is recoverable within the stated constitutive regime

 A2_EQUILIBRIUM:
  Elastic stress fields must satisfy force balance

 A3_COMPATIBILITY:
  Elastic strain fields must correspond to admissible displacement fields

 A4_CONSTITUTIVE_CLOSURE:
  Stress and strain are linked by a declared elastic constitutive law

 A5_OBJECTIVITY:
  Finite elasticity must use objective deformation measures

 A6_BOUNDARY_VALUE_COMPLETENESS:
  Elasticity problems require sufficient boundary conditions for well-posedness

 A7_ENERGY:
  Stable elastic constitutive models admit appropriate positive energy behaviour

 A8_NUMERICAL_CONVERGENCE:
  Discrete elasticity results require mesh and solution convergence checks
}

REASONING {

 R1_ELASTICITY_CONTEXT {
  INPUT = deformable_solid_problem
  IDENTIFY = geometry
  IDENTIFY = material_symmetry
  IDENTIFY = loads_and_temperature
  IDENTIFY = boundary conditions
  IDENTIFY = small_or_finite deformation
  RETURN = elasticity_context
 }

 R2_CONSTITUTIVE_SELECTION {
  INPUT = elasticity_context
  IF isotropic_linear
   SELECT = E_nu_or_Lame parameters
  ELSE
   SELECT = anisotropic_or_nonlinear constitutive tensor/model
  RETURN = elastic_material_model
 }

 R3_EQUILIBRIUM_REASONING {
  INPUT = body_forces_and_boundary tractions
  FORMULATE = div(sigma)+b=0
  RETURN = equilibrium_system
 }

 R4_KINEMATIC_REASONING {
  INPUT = displacement_field
  COMPUTE = infinitesimal_strain_or_finite deformation measure
  CHECK = compatibility
  RETURN = strain_field
 }

 R5_STRESS_RECOVERY {
  INPUT = strain_field_and_constitutive model
  COMPUTE = stress_tensor
  RETURN = elastic_stress_field
 }

 R6_ENERGY_REASONING {
  INPUT = elastic_state
  COMPUTE = strain_energy
  APPLY = minimum_potential_energy_or_virtual work
  RETURN = energetic_solution_profile
 }

 R7_REDUCED_MODEL_REASONING {
  INPUT = geometry_and_constraint assumptions
  SELECT = planeStress_planeStrain_axisymmetric_or_generalisedPlaneStrain
  RETURN = reduced_elasticity_model
 }

 R8_THERMOELASTIC_REASONING {
  INPUT = temperature_field_and_restraints
  COMPUTE = thermal_strain
  COMBINE = mechanical_and_thermal strain
  RETURN = thermoelastic_stress_state
 }

 R9_FINITE_ELEMENT_REASONING {
  INPUT = geometry_material_mesh_loads_boundary data
  FORM = element_stiffness matrices
  ASSEMBLE = global_system
  SOLVE = displacements
  RECOVER = strain_and_stress
  RETURN = finite_element_elasticity solution
 }

 R10_VALIDATION {
  INPUT = elasticity_result
  CHECK = equilibrium_compatibility_boundary conditions_energy positivity_mesh convergence_and_constitutive validity
  RETURN = validated_elasticity_status
 }
}

CONSTRAINTS {
 C1 = "Linear elasticity is invalid when strains, rotations or constitutive nonlinearities are large."
 C2 = "Poisson ratio for stable isotropic linear elasticity lies within the admissible range."
 C3 = "Nearly incompressible elasticity may require special numerical treatment to avoid locking."
 C4 = "Plane stress and plane strain assumptions must not be conflated."
 C5 = "Anisotropic materials require orientation-dependent constitutive data."
 C6 = "Thermal stress requires restraint or incompatible thermal expansion."
 C7 = "Stress singularities may occur at sharp corners, cracks or idealised load points."
 C8 = "Finite element stress recovery can be mesh-sensitive."
 C9 = "Hyperelastic models require objective finite-deformation kinematics."
 C10 = "Boundary conditions must prevent rigid-body modes unless those modes are intended."
}

SEMANTIC_EQUIVALENCE {
 "Elasticity" <=> "ElasticSolidMechanics"
 "LinearElasticity" <=> "HookeanElasticity"
 "ElasticityTensor" <=> "StiffnessTensor"
 "ComplianceTensor" <=> "ElasticComplianceTensor"
 "YoungsModulus" <=> "ElasticModulus"
 "StrainEnergyDensity" <=> "ElasticEnergyDensity"
}

VALIDATION {
 V1 = "Material symmetry and constitutive law are explicit."
 V2 = "Elastic constants are internally consistent."
 V3 = "Small or finite deformation assumptions are documented."
 V4 = "Equilibrium is satisfied."
 V5 = "Compatibility is satisfied."
 V6 = "Boundary conditions are complete and consistent."
 V7 = "Plane stress or plane strain assumptions are justified."
 V8 = "Thermal strain assumptions are explicit."
 V9 = "Finite element solutions are mesh-converged."
 V10 = "Rigid-body modes and singularities are correctly interpreted."
}

CONFIDENCE {
 BASE_CONFIDENCE = 0.99
 ELASTICITY_DEFINITION = 0.99
 CONSTITUTIVE_REASONING = 0.99
 EQUILIBRIUM_REASONING = 0.99
 KINEMATIC_REASONING = 0.99
 STRESS_RECOVERY_REASONING = 0.99
 ENERGY_REASONING = 0.98
 REDUCED_MODEL_REASONING = 0.98
 THERMOELASTIC_REASONING = 0.98
 FINITE_ELEMENT_REASONING = 0.99

 POLICY =
  "Reduce confidence when material symmetry, elastic constants, deformation regime, boundary conditions, thermal data, incompressibility, mesh density or constitutive validity are underspecified."
}

AI_CONSUMPTION {
 PURPOSES = [
  SemanticSearch,
  OntologyGrounding,
  ElasticityReasoning,
  LinearElasticityReasoning,
  NonlinearElasticityReasoning,
  HyperelasticityReasoning,
  HookesLawReasoning,
  ElasticityTensorReasoning,
  PlaneStressReasoning,
  PlaneStrainReasoning,
  ThermoelasticReasoning,
  StrainEnergyReasoning,
  EquilibriumReasoning,
  CompatibilityReasoning,
  FiniteElementElasticityReasoning,
  KnowledgeGraphConstruction,
  RetrievalAugmentation,
  RecursiveInference
 ]

 NORMALISE {
  "elasticity" -> Elasticity
  "linear elasticity" -> LinearElasticity
  "Hooke's law" -> HookesLaw
  "Young's modulus" -> YoungsModulus
  "shear modulus" -> ShearModulus
  "bulk modulus" -> BulkModulus
  "Poisson ratio" -> PoissonRatio
  "elasticity tensor" -> ElasticityTensor
  "strain energy density" -> StrainEnergyDensity
  "hyperelasticity" -> Hyperelasticity
 }

 INTERPRETATION_ORDER = [
  detect_elasticity_context,
  identify geometry_material symmetry_loads_temperature_boundary conditions_and_deformation regime,
  select linear_nonlinear_isotropic_anisotropic_or_hyperelastic constitutive model,
  formulate equilibrium_and_kinematics,
  compute displacement_strain_and_stress fields,
  apply energy_or_variational principles when useful,
  choose planeStress_planeStrain_axisymmetric_or_full3D representation,
  if_numerical solve_with_FEM_or_other method,
  validate equilibrium_compatibility_energy_boundary data_and_convergence,
  return semantic output
 ]
}

EXECUTION_RULES {
 E1:
  IF query_mentions("elasticity")
  OR recoverable_solid_deformation_is_under_analysis
  THEN activate(ElasticityOntology)

 E2:
  IF small_deformation_and_linear_material
  THEN use LinearElasticity

 E3:
  IF isotropic_material
  THEN parameterise_with_two_independent_elastic constants

 E4:
  IF direction_dependent_properties
  THEN use AnisotropicElasticity

 E5:
  IF large_deformation_but_recoverable
  THEN consider FiniteElasticity_or_Hyperelasticity

 E6:
  IF thin_body_with_free_out_of_plane stress
  THEN consider PlaneStress

 E7:
  IF long_or_constrained_body_with_zero_out_of_plane strain
  THEN consider PlaneStrain

 E8:
  IF thermal_loading
  THEN include Thermoelasticity

 E9:
  IF complex_geometry_or_boundary conditions
  THEN consider FiniteElementMethod
}

RECURSION {
 ENTRY = Elasticity

 EXPAND(Elasticity) -> [
  LinearElasticity,
  NonlinearElasticity,
  InfinitesimalElasticity,
  FiniteElasticity,
  Hyperelasticity,
  IsotropicElasticity,
  AnisotropicElasticity,
  OrthotropicElasticity,
  TransverselyIsotropicElasticity,
  ElasticSolid,
  StressTensor,
  StrainTensor,
  DisplacementField,
  DeformationGradient,
  ConstitutiveRelation,
  HookesLaw,
  ElasticityTensor,
  ComplianceTensor,
  YoungsModulus,
  ShearModulus,
  BulkModulus,
  PoissonRatio,
  LameFirstParameter,
  LameSecondParameter,
  StrainEnergyDensity,
  ComplementaryEnergy,
  ElasticPotentialEnergy,
  EquilibriumEquation,
  CompatibilityEquation,
  BoundaryCondition,
  TractionBoundaryCondition,
  DisplacementBoundaryCondition,
  PlaneStress,
  PlaneStrain,
  GeneralisedPlaneStrain,
  SaintVenantPrinciple,
  Torsion,
  Bending,
  AxisymmetricElasticity,
  Thermoelasticity,
  NavierCauchyEquation,
  AiryStressFunction,
  BeltramiMichellEquations,
  VariationalPrinciple,
  PrincipleOfMinimumPotentialEnergy,
  FiniteElementMethod
 ]

 EXPAND(MaterialModels) -> [
  IsotropicElasticity,
  OrthotropicElasticity,
  TransverselyIsotropicElasticity,
  AnisotropicElasticity,
  Hyperelasticity
 ]

 EXPAND(Formulations) -> [
  DisplacementFormulation,
  StressFormulation,
  MixedFormulation,
  VariationalFormulation,
  FiniteElementFormulation
 ]

 STOP_CONDITION =
  "Stop when constitutive law, kinematics, equilibrium, compatibility, boundary conditions, stress-strain field, energy behaviour and numerical validation are sufficiently resolved."
}

ROUTING {
 PARENT_ONTOLOGY = "Engineering Mathematics"

 IMMEDIATE_CONTEXTS = [
  "Engineering Mathematics Ontology",
  "Stress Analysis Ontology",
  "Strain Analysis Ontology",
  "Structural Analysis Ontology"
 ]

 RELATED_ONTOLOGIES = [
  "Engineering Mathematics",
  "Engineering Mathematics Ontology",
  "Stress Analysis Ontology",
  "Strain Analysis Ontology",
  "Structural Analysis Ontology",
  "Solid Mechanics Ontologies",
  "Continuum Mechanics Ontologies",
  "Strength of Materials Ontologies",
  "Tensor Analysis Ontologies",
  "Partial Differential Equation Ontology",
  "Finite Element Ontology",
  "Materials Science Ontologies",
  "Numerical Analysis Ontologies"
 ]

 ROUTE_RULES = [
  "Elasticity -> Engineering Mathematics for parent semantics",
  "Elasticity -> Stress Analysis for stress-state interpretation",
  "Elasticity -> Strain Analysis for deformation measures",
  "Elasticity -> Structural Analysis for member and system response",
  "Elasticity -> Solid and Continuum Mechanics for balance laws",
  "Elasticity -> Tensor Analysis for constitutive tensors and transformations",
  "Elasticity -> PDE Ontology for Navier-Cauchy and elasticity field equations",
  "Elasticity -> Finite Element for complex numerical boundary-value problems",
  "Elasticity -> Materials Science for elastic material properties"
 ]
}

UPDATE_MODEL {
 MODE = "append-compatible semantic versioning"

 MAJOR = "ontology-breaking conceptual change"
 MINOR = "new compatible elastic models, formulations or reasoning pathways"
 PATCH = "clarification, metadata or formatting correction"

 RULES = [
  "Preserve reversible-deformation semantics.",
  "Preserve equilibrium and compatibility semantics.",
  "Preserve stress-strain constitutive semantics.",
  "Preserve isotropic-versus-anisotropic distinction.",
  "Preserve linear-versus-nonlinear distinction.",
  "Preserve plane-stress-versus-plane-strain distinction.",
  "Preserve strain-energy semantics.",
  "Preserve machine-readable elasticity semantics."
 ]
}

OUTPUT {
 CANONICAL_DEFINITION =
  "Elasticity is the mathematical theory of reversible deformation and stress in solids and continua."

 CONSTITUTIVE_SUMMARY =
  "Elastic constitutive laws relate stress and strain through stiffness or compliance descriptions."

 ISOTROPIC_SUMMARY =
  "Isotropic linear elasticity is defined by two independent elastic constants such as Young's modulus and Poisson ratio."

 EQUILIBRIUM_SUMMARY =
  "Elastic stress fields must satisfy equilibrium while strain fields satisfy compatibility."

 ENERGY_SUMMARY =
  "Elastic deformation stores recoverable strain energy."

 FINITE_SUMMARY =
  "Finite elasticity and hyperelasticity extend elastic modelling to large reversible deformation."

 FEM_SUMMARY =
  "Finite element methods approximate elastic boundary-value problems by discretising displacement or mixed fields."

 VALIDATION_SUMMARY =
  "Elasticity solutions should be checked for constitutive validity, equilibrium, compatibility, boundary conditions, energy behaviour and numerical convergence."
}

FLOW {
 INPUT
 -> detect_elasticity_context
 -> identify geometry_material symmetry_elastic constants_loads_temperature_boundary conditions_and_deformation regime
 -> select linear_nonlinear_isotropic_anisotropic_orthotropic_or_hyperelastic model
 -> formulate displacement_kinematics_and_strain measure
 -> apply constitutive relation_to_compute stress
 -> enforce equilibrium_and_compatibility
 -> choose planeStress_planeStrain_axisymmetric_or_full3D formulation
 -> if_thermal include thermal strain_and_thermoelastic coupling
 -> if_energy_method use virtual_work_or_minimum_potential_energy
 -> if_numerical discretise_with_finite_element_or_related method
 -> validate constitutive_consistency_equilibrium_compatibility_boundary data_rigid modes_mesh convergence_and_energy behaviour
 -> route to engineering_mathematics stress_analysis strain_analysis structural_analysis solid_mechanics continuum_mechanics tensor_analysis PDE finite_element materials_science_or_numerical_analysis ontology
 -> recursively_expand_required_dependencies
 -> verify_result
 -> assign_confidence
 -> produce_semantic_output
 -> END
}

SEMANTIC_SUMMARY {
 "The Elasticity Ontology defines reversible deformation of solids and continua. It supports linear and nonlinear elasticity, infinitesimal and finite elasticity, hyperelasticity, isotropic, orthotropic, transversely isotropic and anisotropic materials, stress and strain tensors, displacement and deformation gradients, Hooke's law, stiffness and compliance tensors, Young's, shear and bulk moduli, Poisson ratio and Lamé parameters, strain energy, equilibrium and compatibility, displacement and traction boundaries, plane stress, plane strain, torsion, bending, axisymmetric and thermoelastic formulations, Navier-Cauchy equations, Airy stress functions, Beltrami-Michell equations, variational principles and finite element analysis."
}

FLOW_HASH {
 METHOD = "semantic-content-identity"
 ID = "KIRETH-MATH-APPLIED-MATHEMATICS-ELASTICITY-ONTOLOGY-V1"
}

}

END_FLOW