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**Assumptions introduced to explain a thing must not be multiplied beyond necessity**

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**Basic Terminology • Constitutive Relation: Stress-strain relation**

What is a high temperature problem? Deformation at homologous temperatures above T/Tm >0.35 where Tm is melting temperature. • Stress Relaxation: Decrease in Stress with time at Constant Strain State variables: are variables describing the state of a given thermodynamic system. They include energy, enthalpy, entropy, etc. Plasticity is the propensity of a material to undergo permanent deformation under load • Creep: Increase in Strain with time at Constant Stress Fatigue: failure by repeated stress in materials TMF: is a fatigue of materials where thermal cycles are applied to a test material additional to a mechanical cyclic loading. Damage: the generic name given for material degradation measure Hardening: is an increase in mechanical strength due to plastic deformation

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**Solution of the solid mechanics problem must satisfy:**

Equations of Equilibrium or of motion Relate stress inside a body to body forces and external loads. Three equations relating six stress components. Conditions of geometry or the compatibility of strain and displacement Relate stress inside a body to the displacements Six kinematics equations relating six strain components and three displacements Material constitutive laws or stress-strain relations s=s(e, T, state variables), example: Hooke’s Law Six equations relating stress-strain Additional equations are needed to relate state variables, for example, equations for damage evolution or hardening as functions of stress and strain (strain rate)

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**Example of High Temperature Problem**

High temperature Creep of a Turbine Airfoil There is a noticeable damage - microvoids TMF cracking of a Turbine Airfoil There is a noticeable damage - crack Fractured Tensile Specimen/Test Temperature 2000°F

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**Metal Crystallographic Structure**

BCC <110> <111> <001> <100> FCC HCP

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**PWA 1480 single crystal Elastic Modulus at RT as a function of orientation**

Elastic Behavior <001> <111> 42 msi <100> 18 msi An effect of the anisotropy: Young modulus on the direction of the tensile axis during a tensile test. a) Case of a typical single crystal of Ni-base superalloy. The crystal axes are denoted by [100], [010] and [001]. b) Case of an isotropic material <110> 30 msi <010>

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Deformation Gradient The position (vector) of a particle in the initial, undeformed state of a body is denoted X relative to some coordinate basis. If dX is a line segment joining two nearby particles in the undeformed state and dx is the line segment joining the same two particles in the deformed state, the linear transformation between the two line segments is given by dx=FdX; F - deformation gradient. It is assumed that x is a differentiable function of X and time t, i.e, that cracks and voids do not open or close during the deformation. F is a second-order tensor and contains information about both the stretch and rotation of the body X x Displacement

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**Small vs Large Deformation and Strain**

Change in Length of an element Cauchy-Green strain Displacement gradient D

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**Uniaxial extension of incompressible**

Examples Uniaxial extension of incompressible Simple shear Rotation

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**Principle of Objectivity**

Constitutive equations must be invariant under change of reference. This principal embodies both spatial and temporal reference changes In the formulation of constitutive equations it is necessary to employ variables that are observer independent Distance between two points and the angle between two lines are objective Position and Velocity of a particle will be different in different reference frames Newton’s laws of motion are valid only in special frames; thus they are not objective. Stress rate, strain rate are not objective tensors for arbitrary finite deformation

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Stresses Any of stress tensors can be expressed in any coordinate system as long as you know deformation The 1st Piola-Kirchhoff stress is the 3D generalization of the 1D concept of engineering stress If the material rotates without a change in stress state (rigid rotation), the components of the 2nd Piola-Kirchhoff stress tensor will remain constant, irrespective of material orientation.

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