CE5215-Theory and Applications of Cement Composites_Lecture 6

CE5215-Theory and Applications of Cement Composites

Dr. T. P. Tezeswi Assistant Professor,

Dept. of Civil Engineering NIT-Warangal

Email: tezeswi@nitw.ac.in

Chapter - 2 Stress-Strain Relations

Orthotropic Material • If there are two planes of material property symmetry w.r.to a 3rd mutually

orthogonal plane. • Stress-strain relations in coordinates aligned with principal material

directions are:

– No interaction between normal stresses (σ1, σ2, σ3) and shearing strains γ23, γ31, γ12 – No interaction between shearing stresses and normal stresses – No interaction between shearing stresses and shearing strains in different planes – Only nine independent constants in the stiffness matrix

• Example: Wood. Material properties in three perpendicular directions (axial, radial, and circumferential) are different.

(2.15)

Ref: Mechanics of Composite Materials-Robert M. Jones; Wikipedia

Orthotropic Material • Engg constants (Ei,γij, Gij) are measured in simple uniaxial ,

tension or pure shear tests & have more direct meaning than tensor components

(2.25) Ref: Mechanics of Composite Materials-Robert M. Jones; Wikipedia

Orthotropic Material

Ref: Mechanics of Composite Materials-Robert M. Jones; Wikipedia

Orthotropic Material

Ref: Mechanics of Composite Materials-Robert M. Jones; Wikipedia

Orthotropic Material

Ref: Mechanics of Composite Materials-Robert M. Jones; Wikipedia

Restrictions on Elastic Constants

Ref: Mechanics of Composite Materials-Robert M. Jones; Wikipedia

• For isotropic materials, certain relations between engg. constants must be satisfied. Ex: Shear modulus is defined in terms of E and ν as:

• In order for E & G to be always +ve, i.e., a +ve normal stress or shear stress

times the respective +ve normal strain or shear strain yield +ve work

• If isotropic body is subjected to hydrostatic pressure ρ, then volumetric

strain (= sum of 3 normal or extensional strains) is:

• K is +ve only if E is positive and o

• If bulk modulus (K) is –ve , hydrostatic pressure would cause expansion of a cube of isotropic material.

• For isotropic materials, the Poisson’s ratio is restricted to the range

so that shear or hydrostatic loading does not produce –ve strain energy

(2.38)

(2.39)

Thermodynamic Constraint on Elastic Constants

Ref: Mechanics of Composite Materials-Robert M. Jones

• Orthotropic materials:

• Product of a stress and the corresp. Strain represents work done by the stress.

• Sum of work done by all stress must be positive to avoid creation of energy.

• Matrices relating stress to strain (Cij & Sij) must be positive-definite.

Lamina

Ref: Mechanics of Composite Materials-Robert M. Jones;

• Lamina:

• A flat or curved (shell) arrangement of unidirectional or woven fibers in a supporting matrix.

• Basic building block in laminated fiber composite materials.

• Can withstand high stresses only in the direction of fibers

• In-plane stress is the fundamental capacity.

• Other laminae with different fiber directions help in carrying in-plane stress perpendicular to fibers

Lamina: Various Stress-Strain Behaviors

Ref: Mechanics of Composite Materials-Robert M. Jones

Lamina: Plane-Stress problem

Ref: Mechanics of Composite Materials-Robert M. Jones;

• For a unidirectionally reinforced lamina in the 1-2 plane or woven lamina, a plane state of stress is defined by setting the following in the 3D stress-strain relations:

(2.57,2.58)

Unidirectionally reinforced lamina

Lamina: Plane-Stress problem

Ref: Mechanics of Composite Materials-Robert M. Jones;

Lamina: Plane-Stress problem

Ref: Mechanics of Composite Materials-Robert M. Jones;

Lamina: Plane-Stress problem

Ref: Mechanics of Composite Materials-Robert M. Jones;

Lamina: Plane-Stress problem

Ref: Mechanics of Composite Materials-Robert M. Jones;

Lamina vs Laminate

Ref: Mechanics of Composite Materials-Robert M. Jones

• Laminate: A bonded stack of laminae with various orientations of principle directions

• Layers are usually bonded by the same matrix material used in individual laminae.

Lamina: Plane-Stress problem

Ref: Mechanics of Composite Materials-Robert M. Jones

Lamina: Plane-Stress problem

Ref: Mechanics of Composite Materials-Robert M. Jones

Orthotropic Lamina: Strength Characteristics

Ref: Mechanics of Composite Materials-Robert M. Jones

• Laminate: Because of orthotropy,, axes of principal stress do not coincide with axes of principal strain.

• As strength is lower in one direction, highest stress may not govern design

• Comparison of actual stress field with allowable stress is required

• Stiffness serves as basis for determination of actual stress field.

• To define allowable stress field, establish allowable stresses /strengths in principal material directions.

Orthotropic Lamina: Strength Characteristics

Ref: Mechanics of Composite Materials-Robert M. Jones

• The strengths result from independent application of σ1, σ2, τ12

Orthotropic Lamina: Strength Characteristics

Ref: Mechanics of Composite Materials-Robert M. Jones

• The strengths result from independent application of σ1, σ2, τ12

Mechanical Behavior Of Various Materials

• Anisotropic materials – Application of Normal stress not only causes extension along direction

of stress and contraction in the perpendicular direction, but also shearing deformation.

– Application of Shear stress causes shear deformation, as well as extension and contraction

– Shear-Extension coupling: Coupling between both loading and both deformation modes. Also seen in orthotropic materials subjected to normal stress in a non-principal material direction.

Ref: Mechanics of Composite Materials-Robert M. Jones;