Domain

Remarks

Hooke's Law

• Change in shape & size of a body is proportional to the change in applied loads
• Within the range of "elastic region"
• Clearly illustrated in stress-strain curves

Elastic region

• No permanent deformation suffered
• Linearly elastic: deformed strain directly proportional to applied stress & plot is straight line
• Gradient = Young's modulus, E constant
• Non-linearly elastic: deformed strain unproportional to stress
• E(e ) is function of strain, varies
• Note:
• Both linearly & non-linearly elastic deformation follows their respective stress-strain paths during loading & unloading
• Typical of lightly-loaded homogeneous, isotropic materials

Non-elastic region

• Loading stress-strain path is different from unloading stress-strain path
• Temporary & permanent deformations
• E varies w.r.t. strain & stress path, orientation, rate, etc. à tangent modulus E(s , n , e )
• Hysteresis, phase transition, irregularities, imperfections & other phenomena account for the difference
• Typical of anisotropic, multi-phased materials like soil & polymers

Properties often required for analysis

• E: Young's modulus; within l-elastic region, is ratio of axial stress / axial strain
• n : Poisson's ratio; -ve ratio of lateral strain / longitudinal strain; [0,0.5] à brittle/rigid material: low n à ductile material: high n
• G: shear modulus of elasticity; G=E/2.(1+n ); ratio of shear stress / shear strain
• K: bulk modulus of elasticity; K=E/3.(1-2.n ); ratio of mean normal stress / dilatation (volumetric strain)

Brittle / rigid materials

• Low n
• High E, G
• Low K
• Low ultimate strain

Ductile materials

• High n
• Low E, G
• High K
• High ultimate strain

Stress-strain

• Normal direction: Poisson effect
• e = f(s , E, n )
• Shear direction: no Poisson effect
• g = f(t , G)

Implications on normal stress-strain

• Normal strain can be induced, under zero stress in that direction
• Due to Poisson's ratio
• Normal stresses & strains cause change in size & volume
• No failure: equal triaxial normal stresses
• In linearly-elastic region, g _xy = 2.e

Implications on shear stress-strain

• Shear strain is only influenced by shear stress; no Poisson's ratio
• Only shear strain is induced by shear stress in that direction
• Shear stresses & strains cause change in shape (distortion) only
• Failure: under unequal triaxial normal stresses, causing shear distortion

Elasticity relation

• {s } = {D} . {e }
• Poisson effect: stress & strain are coupled

State of stress-strain

• State of stress: complete description of stresses at a point à using consistent coordinate system: Cartesian, polar, cylindrical
• Plane stress: no out-of-plane stress; e.g. on free surface
• State of strain: complete description of strains at a point
• Plane strain: no out-of-plane strain; e.g. planes across straight, long, prismatic, homogeneous structures
• Axisymmetric: stress & strain invariant with angle of revolution of a pressure cylinder
• Poisson effect: normal in-plane & out-of-plane strains coupled à plane stress cannot occur simultaneously with plane strain

Stress transformation

• Equilibrium
• State of stress at a point varies with {loads, locations, time, orientation planes}
• Principal stress: max. stress at point orientation
• Principal normal stress p.n.s.: zero shear stress
• Principal shear stress: 45° to p.n.s.; av. n.s.
• Retrofit or reinforce along principal stress directions
• Mohr’s circle

Strain transformation

• Compatibility
• State of strain at a point varies with {geometry, locations, time, orientation planes}
• Principal strain: max. stress at point orientation
• Principal normal strain p.n.s.: zero shear strain
• Principal shear strain: 45° to p.n.s.; av. n.s.
• Restrain along principal strain directions
• Mohr’s circle

3-D absolute max shear

• Occurs in-plane: if principal stresses or strains are opposite à tension-compression or compression-tension
• Occurs out-of-plane: if principal stresses or strains are same à tension-tension or compression-compression