Stress Analysis Ontology

AI INTERPRETATION NOTE
This Flow Algebra defines the Stress Analysis Ontology within the Kireth Research Semantic Library. Stress analysis studies the distribution, transformation and consequences of internal forces within solids and structures under mechanical, thermal and other loads. This ontology supports normal and shear stress, stress tensors, principal stresses, invariants, Mohr's circle, plane stress, plane strain, hydrostatic and deviatoric stress, von Mises and Tresca measures, stress concentration, contact stress, thermal stress, elastic stress fields, failure criteria, finite element stress recovery and relationships with structural analysis, solid mechanics, elasticity, materials and numerical analysis.

FLOW_ALGEBRA_RECORD {

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

IDENTITY {
 PRIMARY_ENTITY = StressAnalysis
 SYMBOLIC_FORMS = [
  "sigma = F/A",
  "tau = V/A",
  "t(n)=sigma_tensor*n",
  "det(sigma_tensor-lambda I)=0"
 ]
 ENTITY_CLASS = ContinuumStressEvaluationFramework
 PRIMARY_CONTEXT = EngineeringMathematics
 DEFINITION = "Stress analysis is the mathematical determination of internal stress states, stress transformations and stress-based response measures in loaded solids and structures."
 SEMANTIC_ROLE = "Provides machine-readable semantics for stress tensors, normal and shear stress, principal stresses, invariants, stress transformation, concentration effects, failure measures and numerical stress recovery."
}

SCOPE {
 INCLUDE = [
  StressAnalysis,
  Stress,
  StressTensor,
  CauchyStressTensor,
  NormalStress,
  ShearStress,
  PrincipalStress,
  PrincipalDirection,
  MaximumShearStress,
  MeanStress,
  HydrostaticStress,
  DeviatoricStress,
  StressInvariant,
  FirstInvariant,
  SecondInvariant,
  ThirdInvariant,
  StressTransformation,
  MohrsCircle,
  PlaneStress,
  PlaneStrain,
  ThreeDimensionalStress,
  UniaxialStress,
  BiaxialStress,
  MultiaxialStress,
  AxialStress,
  BendingStress,
  TorsionalShearStress,
  TransverseShearStress,
  BearingStress,
  ContactStress,
  HertzianContactStress,
  ThermalStress,
  ResidualStress,
  StressConcentration,
  StressConcentrationFactor,
  NotchStress,
  SingularStress,
  EquivalentStress,
  VonMisesStress,
  TrescaStress,
  YieldCriterion,
  FailureCriterion,
  ElasticStressField,
  ConstitutiveRelation,
  Strain,
  HookesLaw,
  FiniteElementStress,
  StressRecovery,
  NodalStress,
  GaussPointStress
 ]

 EXCLUDE_AS_PRIMARY = [
  StrainAnalysisWithoutStressInterpretation,
  PureStructuralForceAnalysisWithoutStressRecovery,
  FluidStressWithoutSolidMechanicsContext,
  FatigueLifePredictionWithoutStressStateAsPrimaryContext
 ]

 DISAMBIGUATION_RULE =
  "When internal force intensity, stress tensor state, stress transformation, stress concentration or stress-based failure measure is being evaluated in a solid or structure, interpret the context as stress analysis."
}

CORE_CONCEPTS {

 StressAnalysis {
  definition = "The mathematical evaluation of stress states in solids and structural components."
 }

 StressTensor {
  definition = "A second-order tensor describing normal and shear stress components acting through a point."
 }

 NormalStress {
  definition = "Stress acting perpendicular to a material plane."
 }

 ShearStress {
  definition = "Stress acting tangentially to a material plane."
 }

 PrincipalStress {
  definition = "A normal stress acting on a plane where shear stress is zero."
 }

 DeviatoricStress {
  definition = "The stress component remaining after subtraction of mean hydrostatic stress."
 }

 HydrostaticStress {
  definition = "The isotropic mean-stress component of the stress tensor."
 }

 StressInvariant {
  definition = "A scalar quantity derived from the stress tensor that is independent of coordinate orientation."
 }

 StressConcentration {
  definition = "A local amplification of stress caused by geometry, discontinuity or load introduction."
 }

 EquivalentStress {
  definition = "A scalar stress measure used to compare multiaxial stress states with material failure criteria."
 }
}

RELATIONS {
 StressAnalysis --is_child_of--> EngineeringMathematics
 StressTensor --contains--> NormalStress
 StressTensor --contains--> ShearStress
 CauchyStressTensor --maps--> SurfaceTraction
 PrincipalStress --is_eigenvalue_of--> StressTensor
 PrincipalDirection --is_eigenvector_of--> StressTensor
 HydrostaticStress --is_mean_part_of--> StressTensor
 DeviatoricStress --is_remaining_part_of--> StressTensor
 StressInvariant --is_coordinate_independent_property_of--> StressTensor
 MohrsCircle --represents--> StressTransformation
 PlaneStress --is_reduction_of--> ThreeDimensionalStress
 PlaneStrain --constrains--> StrainState
 AxialStress --results_from--> AxialForce
 BendingStress --results_from--> BendingMoment
 TorsionalShearStress --results_from--> Torsion
 StressConcentrationFactor --quantifies--> StressConcentration
 VonMisesStress --depends_on--> DeviatoricStress
 TrescaStress --depends_on--> PrincipalStressDifferences
 FiniteElementStress --derived_from--> FiniteElementDisplacement
}

OPERATORS {
 DIRECT_NORMAL_STRESS(F,A) :=
  sigma=F/A

 DIRECT_SHEAR_STRESS(V,A) :=
  tau=V/A

 CAUCHY_TRACTION(sigma_tensor,n) :=
  t=sigma_tensor*n

 PRINCIPAL_STRESS_PROBLEM(sigma_tensor) :=
  det(sigma_tensor-lambda*I)=0

 MEAN_STRESS(sigma1,sigma2,sigma3) :=
  sigma_m=(sigma1+sigma2+sigma3)/3

 DEVIATORIC_STRESS(sigma_tensor) :=
  s=sigma_tensor-sigma_m*I

 VON_MISES_STRESS(sigma1,sigma2,sigma3) :=
  sigma_vm=sqrt(((sigma1-sigma2)^2+(sigma2-sigma3)^2+(sigma3-sigma1)^2)/2)

 TRESCA_STRESS(sigma1,sigma2,sigma3) :=
  sigma_T=max(abs(sigma1-sigma2),abs(sigma2-sigma3),abs(sigma3-sigma1))

 PLANE_STRESS_TRANSFORMATION(sigma_x,sigma_y,tau_xy,theta) :=
  sigma_x_prime=(sigma_x+sigma_y)/2
  +(sigma_x-sigma_y)/2*cos(2*theta)
  +tau_xy*sin(2*theta)

 MAX_PLANE_SHEAR(sigma_x,sigma_y,tau_xy) :=
  tau_max=sqrt(((sigma_x-sigma_y)/2)^2+tau_xy^2)

 BENDING_STRESS(M,y,I) :=
  sigma=-M*y/I

 TORSION_CIRCULAR(T,r,J) :=
  tau=T*r/J

 STRESS_CONCENTRATION(Kt,sigma_nominal) :=
  sigma_max=Kt*sigma_nominal
}

DEFINITIONS {
 D1 = "Stress is force intensity acting internally across imagined material surfaces."
 D2 = "The Cauchy stress tensor fully defines traction on any plane through a material point."
 D3 = "Principal stresses are stress-tensor eigenvalues."
 D4 = "Principal directions are mutually orthogonal for a symmetric stress tensor."
 D5 = "Hydrostatic stress changes mean pressure-like stress without contributing to distortional von Mises stress."
 D6 = "Deviatoric stress governs distortion-related yielding measures."
 D7 = "Mohr's circle provides a graphical representation of two-dimensional stress transformation."
 D8 = "Stress concentrations amplify local stress above nominal values."
 D9 = "Equivalent stresses reduce multiaxial states to scalar failure measures."
 D10 = "Finite element stress values depend on displacement interpolation and stress recovery."
}

AXIOMS {
 A1_CAUCHY_STRESS:
  Local traction on any plane is determined by the Cauchy stress tensor

 A2_STRESS_SYMMETRY:
  In classical continua without couple stresses the Cauchy stress tensor is symmetric

 A3_COORDINATE_TRANSFORMATION:
  Stress components transform according to tensor transformation laws

 A4_PRINCIPAL_STRESSES:
  Principal stresses are invariant eigenvalues of the stress tensor

 A5_EQUILIBRIUM:
  Stress fields must satisfy local or global equilibrium with applied loads

 A6_CONSTITUTIVE_COUPLING:
  Stress depends on strain through the declared material constitutive model

 A7_FAILURE_MEASURE_DEPENDENCE:
  Equivalent-stress criteria depend on material behaviour and failure assumptions

 A8_NUMERICAL_RECOVERY:
  Discrete stress results depend on element formulation, mesh quality and recovery method
}

REASONING {

 R1_STRESS_CONTEXT {
  INPUT = loaded_solid_or_structure
  IDENTIFY = geometry
  IDENTIFY = loads
  IDENTIFY = material_model
  IDENTIFY = stress_state_dimension
  RETURN = stress_context
 }

 R2_STRESS_COMPONENTS {
  INPUT = internal_forces_or_field solution
  COMPUTE = normal_and_shear stress components
  RETURN = local_stress_tensor
 }

 R3_STRESS_TRANSFORMATION {
  INPUT = stress_tensor_and_orientation
  TRANSFORM = components_to_new coordinate system
  RETURN = transformed_stress
 }

 R4_PRINCIPAL_STRESS_REASONING {
  INPUT = stress_tensor
  COMPUTE = eigenvalues_and_eigenvectors
  RETURN = principal_stresses_and_directions
 }

 R5_INVARIANT_REASONING {
  INPUT = stress_tensor
  COMPUTE = trace_determinant_and_deviatoric invariants
  RETURN = coordinate_invariant_stress measures
 }

 R6_FAILURE_MEASURE_REASONING {
  INPUT = principal_or_tensor stress_and_material type
  SELECT = vonMises_Tresca_or_other criterion
  COMPUTE = equivalent_stress
  RETURN = failure_measure_profile
 }

 R7_STRESS_CONCENTRATION_REASONING {
  INPUT = nominal_stress_and_geometry discontinuity
  DETERMINE = Kt_or_local_field
  COMPUTE = peak_stress
  RETURN = concentration_profile
 }

 R8_THERMAL_STRESS_REASONING {
  INPUT = temperature_change_constraints_material properties
  COMPUTE = constrained_thermal_strain_and_stress
  RETURN = thermal_stress_profile
 }

 R9_FEA_STRESS_REASONING {
  INPUT = finite_element displacement field
  COMPUTE = strains_at_integration points
  APPLY = constitutive_matrix
  RECOVER = element_nodal_or_smoothed stresses
  RETURN = finite_element_stress field
 }

 R10_VALIDATION {
  INPUT = stress_result
  CHECK = equilibrium_boundary tractions_constitutive consistency_mesh convergence_and_singularity effects
  RETURN = validated_stress_status
 }
}

CONSTRAINTS {
 C1 = "Nominal stress formulas may be invalid near geometric discontinuities."
 C2 = "Plane stress and plane strain assumptions must not be conflated."
 C3 = "Stress singularities do not generally converge to finite point values under mesh refinement."
 C4 = "Von Mises and Tresca criteria are mainly suited to ductile yielding interpretations."
 C5 = "Brittle failure may require principal-stress or other material-specific criteria."
 C6 = "Contact stresses require appropriate contact geometry and constitutive assumptions."
 C7 = "Thermal stress requires restraint or nonuniform thermal expansion."
 C8 = "Finite element nodal stress smoothing can obscure local discontinuities."
 C9 = "Peak finite element stress should be interpreted with mesh and singularity awareness."
 C10 = "Residual stress can exist without current external loading."
}

SEMANTIC_EQUIVALENCE {
 "StressAnalysis" <=> "MechanicalStressAnalysis"
 "CauchyStressTensor" <=> "StressTensor"
 "PrincipalStress" <=> "PrincipalNormalStress"
 "VonMisesStress" <=> "DistortionEnergyEquivalentStress"
 "TrescaStress" <=> "MaximumShearEquivalentStress"
 "StressConcentrationFactor" <=> "Kt"
}

VALIDATION {
 V1 = "Load and geometry definitions are explicit."
 V2 = "Stress-state dimensional assumptions are documented."
 V3 = "Stress tensor symmetry is checked where classical continuum assumptions apply."
 V4 = "Principal stresses match tensor invariants."
 V5 = "Stress transformations are coordinate-consistent."
 V6 = "Failure criterion matches material behaviour."
 V7 = "Stress concentration interpretation distinguishes nominal and local stress."
 V8 = "Thermal restraint assumptions are explicit."
 V9 = "Finite element stress results are mesh-checked."
 V10 = "Singularities are distinguished from physically finite stress concentrations."
}

CONFIDENCE {
 BASE_CONFIDENCE = 0.99
 STRESS_ANALYSIS_DEFINITION = 0.99
 COMPONENT_REASONING = 0.99
 TRANSFORMATION_REASONING = 0.99
 PRINCIPAL_STRESS_REASONING = 0.99
 INVARIANT_REASONING = 0.99
 FAILURE_MEASURE_REASONING = 0.98
 CONCENTRATION_REASONING = 0.98
 THERMAL_STRESS_REASONING = 0.98
 FEA_STRESS_REASONING = 0.98

 POLICY =
  "Reduce confidence when geometry, loading, stress-state assumptions, material model, failure criterion, contact conditions, thermal restraints, mesh density or singularity behaviour are underspecified."
}

AI_CONSUMPTION {
 PURPOSES = [
  SemanticSearch,
  OntologyGrounding,
  StressAnalysisReasoning,
  StressTensorReasoning,
  PrincipalStressReasoning,
  StressTransformationReasoning,
  MohrsCircleReasoning,
  StressInvariantReasoning,
  PlaneStressReasoning,
  PlaneStrainReasoning,
  BendingStressReasoning,
  TorsionalStressReasoning,
  StressConcentrationReasoning,
  ThermalStressReasoning,
  VonMisesReasoning,
  TrescaReasoning,
  ContactStressReasoning,
  FiniteElementStressReasoning,
  KnowledgeGraphConstruction,
  RetrievalAugmentation,
  RecursiveInference
 ]

 NORMALISE {
  "stress analysis" -> StressAnalysis
  "stress tensor" -> StressTensor
  "Cauchy stress tensor" -> CauchyStressTensor
  "principal stress" -> PrincipalStress
  "Mohr's circle" -> MohrsCircle
  "von Mises stress" -> VonMisesStress
  "Tresca stress" -> TrescaStress
  "stress concentration factor" -> StressConcentrationFactor
  "Kt" -> StressConcentrationFactor
  "thermal stress" -> ThermalStress
 }

 INTERPRETATION_ORDER = [
  detect_stress_analysis_context,
  identify geometry_loads_material_model_and_stress_state assumptions,
  compute local_normal_shear_or_tensor stress,
  transform stress_if_orientation changes,
  compute principal_stresses_directions_and_invariants,
  evaluate hydrostatic_deviatoric_and_equivalent stress measures,
  assess concentrations_contact_or_thermal stress when relevant,
  if_FEA recover_stress_from_displacement field,
  validate equilibrium_material criterion_mesh behaviour_and_singularities,
  return semantic output
 ]
}

EXECUTION_RULES {
 E1:
  IF query_mentions("stress analysis")
  OR solid_internal_stress_state_is_under_analysis
  THEN activate(StressAnalysisOntology)

 E2:
  IF two_dimensional_thin_body
  THEN consider PlaneStress

 E3:
  IF long_constrained_body_with_negligible_out_of_plane strain
  THEN consider PlaneStrain

 E4:
  IF stress_tensor_given
  THEN compute principal stresses_and_invariants

 E5:
  IF ductile_yielding assessment
  THEN consider VonMises_or_Tresca criterion

 E6:
  IF geometric_discontinuity_present
  THEN distinguish nominal_stress_from_stress concentration

 E7:
  IF thermal_change_and_restraint present
  THEN compute ThermalStress

 E8:
  IF contact_loading present
  THEN route to ContactStress reasoning

 E9:
  IF finite_element result
  THEN assess integration_point_stress_mesh convergence_and_singularity
}

RECURSION {
 ENTRY = StressAnalysis

 EXPAND(StressAnalysis) -> [
  Stress,
  StressTensor,
  CauchyStressTensor,
  NormalStress,
  ShearStress,
  PrincipalStress,
  PrincipalDirection,
  MaximumShearStress,
  MeanStress,
  HydrostaticStress,
  DeviatoricStress,
  StressInvariant,
  StressTransformation,
  MohrsCircle,
  PlaneStress,
  PlaneStrain,
  ThreeDimensionalStress,
  UniaxialStress,
  BiaxialStress,
  MultiaxialStress,
  AxialStress,
  BendingStress,
  TorsionalShearStress,
  TransverseShearStress,
  BearingStress,
  ContactStress,
  HertzianContactStress,
  ThermalStress,
  ResidualStress,
  StressConcentration,
  StressConcentrationFactor,
  NotchStress,
  SingularStress,
  EquivalentStress,
  VonMisesStress,
  TrescaStress,
  YieldCriterion,
  FailureCriterion,
  ElasticStressField,
  ConstitutiveRelation,
  Strain,
  HookesLaw,
  FiniteElementStress,
  StressRecovery
 ]

 EXPAND(StressMeasures) -> [
  PrincipalStress,
  MaximumShearStress,
  HydrostaticStress,
  DeviatoricStress,
  VonMisesStress,
  TrescaStress
 ]

 EXPAND(LocalEffects) -> [
  StressConcentration,
  NotchStress,
  ContactStress,
  HertzianContactStress,
  ThermalStress,
  ResidualStress,
  SingularStress
 ]

 STOP_CONDITION =
  "Stop when the local stress tensor, transformed and principal stresses, invariants, equivalent measures, concentration effects, material criterion and numerical validation are sufficiently resolved."
}

ROUTING {
 PARENT_ONTOLOGY = "Engineering Mathematics"

 IMMEDIATE_CONTEXTS = [
  "Engineering Mathematics Ontology",
  "Structural Analysis Ontology",
  "Solid Mechanics Ontologies",
  "Elasticity Ontologies"
 ]

 RELATED_ONTOLOGIES = [
  "Engineering Mathematics",
  "Engineering Mathematics Ontology",
  "Structural Analysis Ontology",
  "Solid Mechanics Ontologies",
  "Elasticity Ontologies",
  "Strength of Materials Ontologies",
  "Strain Analysis Ontologies",
  "Failure Theory Ontologies",
  "Contact Mechanics Ontologies",
  "Finite Element Ontology",
  "Tensor Analysis Ontologies",
  "Materials Science Ontologies",
  "Numerical Analysis Ontologies"
 ]

 ROUTE_RULES = [
  "Stress Analysis -> Engineering Mathematics for parent semantics",
  "Stress Analysis -> Structural Analysis for member forces and global response",
  "Stress Analysis -> Solid Mechanics for continuum balance and constitutive laws",
  "Stress Analysis -> Elasticity for elastic stress fields",
  "Stress Analysis -> Strain Analysis for stress-strain coupling",
  "Stress Analysis -> Failure Theory for yield and fracture criteria",
  "Stress Analysis -> Contact Mechanics for Hertzian and interface stresses",
  "Stress Analysis -> Tensor Analysis for stress transformation and invariants",
  "Stress Analysis -> Finite Element for numerical stress recovery",
  "Stress Analysis -> Materials Science for material-specific failure interpretation"
 ]
}

UPDATE_MODEL {
 MODE = "append-compatible semantic versioning"

 MAJOR = "ontology-breaking conceptual change"
 MINOR = "new compatible stress measures, failure criteria or reasoning pathways"
 PATCH = "clarification, metadata or formatting correction"

 RULES = [
  "Preserve tensor-stress semantics.",
  "Preserve normal-versus-shear distinction.",
  "Preserve principal-stress semantics.",
  "Preserve hydrostatic-versus-deviatoric decomposition.",
  "Preserve plane-stress-versus-plane-strain distinction.",
  "Preserve nominal-versus-local stress distinction.",
  "Preserve failure-criterion context.",
  "Preserve machine-readable stress-analysis semantics."
 ]
}

OUTPUT {
 CANONICAL_DEFINITION =
  "Stress analysis determines internal stress states, their transformations and their implications within loaded solids and structures."

 TENSOR_SUMMARY =
  "The Cauchy stress tensor defines traction on every plane through a material point."

 PRINCIPAL_SUMMARY =
  "Principal stresses are the eigenvalues of the stress tensor and act on planes of zero shear."

 INVARIANT_SUMMARY =
  "Stress invariants provide coordinate-independent descriptions of stress state."

 FAILURE_SUMMARY =
  "Von Mises and Tresca equivalent stresses provide common ductile-yield measures for multiaxial stress states."

 CONCENTRATION_SUMMARY =
  "Geometric discontinuities can amplify local stress above nominal values."

 THERMAL_SUMMARY =
  "Thermal stresses develop when thermal expansion is restrained or spatially nonuniform."

 FEA_SUMMARY =
  "Finite element stresses are recovered from displacement-derived strain and constitutive relations."

 VALIDATION_SUMMARY =
  "Stress-analysis results should be checked for equilibrium, transformation consistency, material applicability, mesh convergence and singularity effects."
}

FLOW {
 INPUT
 -> detect_stress_analysis_context
 -> identify geometry_loads_material_properties_constraints_and_stress_state dimension
 -> derive nominal_or_continuum stress components
 -> construct stress_tensor
 -> if_required transform stress_to_new_orientation
 -> compute principal_stresses_principal_directions_and_invariants
 -> decompose hydrostatic_and_deviatoric stress
 -> compute vonMises_Tresca_or_other equivalent measure as appropriate
 -> assess bending_torsion_contact_thermal_residual_or_concentration effects
 -> if_finiteElement recover_and_interpret integration-point_or_nodal stresses
 -> validate equilibrium_boundary tractions_constitutive assumptions_mesh convergence_and_singularity behaviour
 -> route to engineering_mathematics structural_analysis solid_mechanics elasticity strain_analysis failure_theory contact_mechanics tensor_analysis finite_element materials_science_or_numerical_analysis ontology
 -> recursively_expand_required_dependencies
 -> verify_result
 -> assign_confidence
 -> produce_semantic_output
 -> END
}

SEMANTIC_SUMMARY {
 "The Stress Analysis Ontology defines internal stress evaluation in solids and structures. It supports Cauchy stress tensors, normal and shear stresses, principal stresses and directions, maximum shear, hydrostatic and deviatoric stress, tensor invariants, stress transformations and Mohr's circle, plane stress and plane strain, axial, bending, torsional, transverse-shear and bearing stresses, contact and Hertzian stress, thermal and residual stress, stress concentration and singularity, von Mises and Tresca equivalent stresses, yield and failure criteria, constitutive relations and finite element stress recovery."
}

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

}

END_FLOW