Tensile Testing 1

Tensile Testing

Plane stress and Plane strain • Plane stress and Plane strain • Plane Stress : State of stress in which one or two of the pairs of faces on an elemental • cube are free of any stresses. e.g Torsion of thin wall tube. Expansion of a thin walled • spherical shell under internal pressure. • Plane strain: State of tress where one of the pair of faces on an elemental undergoes • zero strain. e.g Torsion of thin walled tube. Piece of material being compressed in a die .

Mechanical properties The mechanical properties of a material are obtained by subjecting a specimen to prescribed loads and then measuring the resulting deformation. Usually, the test is carried out on a special machine that is specifically designed for this purpose. The measurements of the load and of the deformation are carried out .

Tensile Test Machine (Instron)

Electronic Extensometer

Extensometer

Grip Specimen Grip

Tensile Testing (stretching or pulling)

Determine the behavior of the engineering material under load

Tensile Test -Tensile test determines the strength of the material when subjected to a simple stretching operation. -Standard dimension test samples are pulled slowly and at uniform rate in a testing machin. -The strain ( the elongation of the sample) is defined as:  Engineering Strain = ε ε = (change in length)/(original length) = δ/L0 -The stress ( the applied force divided by the original crosssectional area) is defined as:  Engineering Stress = σ σ = (applied force)/(original area) =P/A0

Stress-Strain Curve

. • Elastic Limit: Greatest stress a material is capable of developing without a permanent set. • Note; elastic limit for metals do not differ widely from the values of the proportionality. • Elastic limit may be taken as that stress at which there is a permanent set of 0.2%.It is therefore higher than limit of proportionality. (suggested by some authors) • Hooke s’ law: Stress is directly proportional to strain in the elastic range. • Young s’ Modulus: It is ratio between stress and reversible strain (stiffness) • It is in fact a measure of the inter atomic bonding forces.

Yield strength/ Proof stress: usually defined as the stress which produces a measurable amount of permanent strain i.e. 0.2% or 0.1%.

Tensile strength - the maximum stress applied to the specimen.

Failure stress - the stress applied to the specimen at failure (usually less than the maximum tensile strength because necking reduces the cross-sectional area)

Ductility % Elongation: % elongation is a measure of ductility, which is given by: % elongation =100 * (Lo - Lf)/ Lo where, Lo = Initial length Lf = Final Length

Ductility % Reduction in Area: % reduction in area is a measure of ductility, which is given by: % reduction in area =100 * (Ao - Af)/ Ao where, Ao = Initial area Af = Final area

Poisson’s ratio When pulled in tension (X), a sample gets longer and thinner, i.e., a contraction in the width (Y) and breadth (Z) Poisson’s ratio: when strained in the (X) direction how much strain occurs in the lateral directions (Y & Z) For most metals this value is 0.333

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This is the Poisson effect. Poisson’s ratio, which is the negative ratio of the contraction over the extension, is also an elastic constant. Poisson's Ratio, laterial strain longitudinal strain ε ν ε =−=−⊥ x unloaded y z loaded P P •Metals: ν = 0.3 – 0.35 •Ceramics: ν = 0.2 – 0.25 •Polymers: ν = 0.25 – 0. 5

Modulus of elasticity - the initial slope of the curve, related directly to the strength of the atomic bonds.

• Resilience: The ability of a material to absorb energy with in elastic limit. • Measure by the modulus of resilience. -Which is strain energy per unit volume. -Stress the material from zero stress to the yield stress. • Energy: Force multiplied by the distance over which it acts.

Modulus of resilience - the area under the linear part of the curve, measuring the stored elastic energy.

Toughness The ability of a material to absorb energy in the plastic range.

Toughness - The total area under the curve, which measures the energy absorbed by the specimen in the process of breaking.

Stress-Strain Curve

True Stress and True Strain • True stress, σ, is the load,P, divided by the instantaneous area of the specimen, Ai. • True Strain: Change in gage length with respect to the instantaneous gage length over which the change occurs. • Up to strain where necking begins, specimen deforms with a constant volume in gauge section. • Constant Volume gives: Ao Lo = Ai Li

True Stress

P Ao Lo σ = ; Ai = Ai Li PLi σ= ; Ao Lo P S= ; Ao Li = Lo + ∆L;

Li Lo + ∆L = = (1 + e) Lo Lo

σ = S (1 + e)

Assumes constant volume. Valid for all strains up to  point where necking  begins. .

True Strain: • True Strain: Change in gage length with respect to the instantaneous gage length over which the change occurs. True strain, ε, is determined from the rate of change in gauge length with respect to the instantaneous gauge length, Li.

True Strain dLi dε = Li  Li  ε = ln L    o Li = Lo +∆L;

ε = ln(1 +e)

Assumes constant volume. Valid for all strains up to point where necking begins.

Engineering Vs. True Stress-Strain Curves Stress

True Stress ­ Strain  Curve

Fracture

Ultimate Tensile Strength

Fracture

Engineering   Stress ­ Strain   Curve

Strain

Holloman Petch Relation ship An empirical relationship was proposed by Holloman in 1945 to describe the shape of the stress-strain curve.

σ = Kεn • σ true stress, ε is true strain, K is strength coefficient, and n is the strain-hardening exponent. • Thus, one can obtain n from a log-log plot of σ versus ε. K is the true stress at ε = 1.0.

• n = 0 for perfectly plastic solids • n = 1 for perfectly elastic solids • n = 0.1 – 0.5 for most metals

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• IN Stress-Strain Curves • Plastic deformation is uniform and permanent between the elastic limit and the UTS. • • Plastic deformation becomes nonuniform once the UTS is exceeded. • In tension this non-uniform deformation manifests itself • as “necking”

Criterion for necking Increase in true stress (due to reduction in cross-sectional area) as the specimen elongates is more than to load carrying capacity due to strain hardening.

Criteria for Instability

Rate of Geometrical Softening and Rate of Work Hardening

Holloman Petch Relation ship An empirical relationship was proposed by Holloman in 1945 to describe the shape of the stress-strain curve.

σ = Kεn • σ true stress, ε is true strain, K is strength coefficient, and n is the strain-hardening exponent. • Thus, one can obtain n from a log-log plot of σ versus ε. K is the true stress at ε = 1.0.

• n = 0 for perfectly plastic solids • n = 1 for perfectly elastic solids • n = 0.1 – 0.5 for most metals

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Uniform Plastic Flow • The stress-strain curve (i.e., flow curve) in the region of uniform plastic deformation does not increase proportionally with strain. The material is said to work harden (or strain harden). • An empirical mathematical relationship was advanced by Holloman in 1945 to describe the shape of the engineering stress-strain curve. σ = Kεn, where is the σ true stress, ε is true strain, K is strength coefficient, and n is the strain-hardening exponent. Thus, one can obtain n from a log-log plot of σ versus ε. K is the true stress at ε = 1.0.

Criteria for Necking

0.2 % Offset Yield StrengthOffset Yield Strength Defining the yield stress as the point separating elastic from plastic deformation is easier than determining that point. The elastic portion of the curve is not perfectly linear, and microscopic amounts of deformation can occur. As a matter of practical convenience, the yield strength is determined by constructing a line parallel to the initial portion of the stress-strain curve but offset by 0.2% from the origin. The intersection of this line and the measured stress-strain line is used as an approximation of the material's yield strength, called the 0.2% offset yield.

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Stress-Strain Curves • Plastic deformation is uniform and permanent between the elastic limit and the UTS. • Plastic deformation becomes non-uniform once the UTS is exceeded. In tension this non-uniform deformation manifests itself as “necking” Uniform plastic strain Non-uniform plastic strain L3 2 Page 47 Uniform Plastic Flow • The stress-strain curve (i.e., flow curve) in the region of uniform plastic deformation does not increase proportionally with strain. The material is said to work harden (or strain harden). • An empirical mathematical relationship was advanced by Holloman in 1945 to describe the shape of the engineering stress-strain curve. σ = Kεn, where is the σ true stress, ε is true strain, K is strength coefficient, and n is the strain-hardening exponent. Thus, one can obtain n from a log-log plot of σ versus ε. K is the true stress at ε = 1.0. Page 48

• True strain, ε, is determined from the rate of change in gauge length with respect to the instantaneous gauge length, Li. • Up to strain where necking begins, specimen deforms with a constant volume in gauge section. • Constant Volume gives:

True Stress

P Ao Lo σ = ; Ai = Ai Li PLi σ= ; Ao Lo P S= ; Ao Li = Lo + ∆L;

Li Lo + ∆L = = (1 + e) Lo Lo

σ = S (1 + e)

Assumes constant volume. Valid for all strains up to  point where necking  begins; Hence, valid for S < Su. Special Case,  True Fracture Stress:

σf=

Pf

Af

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Stress: The true stress is defined as the ratio of the applied load to the

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True stress: can be related to the engineering stress if we assume that there is

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instantaneous cross-sectional area;

no volume change in the specimen. Under this assumption, which leads to http://ceeweb.egr.duke.edu/~dolbow/TENSILE/tutorial/img11.png

True Strain dLi dε = Li

Assumes constant volume. Valid for all strains up to point where necking begins; Hence, valid for S < Su.

 Li  ε = ln    Lo  Li = Lo + ∆L;

ε = ln(1 + e)

Special Case,  True Fracture Strain:

 Ao  ε f = ln  A   f

True Strain: • True Strain: Change in gage length with respect to the instantaneous gage length over which the change occurs.

• εtrue = ln(1 + εe). • σtrue = (1 + εe)(σe),

0.2 % Offset Yield Strength