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Understand the Tensile Stress Strain Curve in One Article

The properties of a material in terms of deformation and damage under tensile forces can be measured by the tensile stress-strain curve, which is one of the most fundamental and important concepts in the mechanics of materials, come with me to understand it.

tensile stress-strain

The horizontal coordinate of the curve is the strain and the vertical coordinate is the stress. The shape of the curve reflects the various deformation processes that occur in the material under the action of external forces.

★ What is Tensile Stress-Strain?

We all know that the three elements of force are magnitude, direction, and point of action. However, the point of action does not have dimensions, it simply represents the location of the force. The material of an object has dimensions and when we need to study the forces at various points within an object, we need to introduce the concept of stress, expressed as σ.

Tensile stress formula: σ= dF/dA, which represents the stress per unit area inside the material. In layman’s terms, tensile stress is the resistance per unit area inside the object when the object is subjected to external action, with a sense of common defence against external enemies.


As shown above, an object is subjected to a tensile force, then in order to balance the force, the material per unit area inside the object is subjected to a portion of the force. When a plane inside the object is perpendicular to the direction of the force and the material is uniform, the average tensile stress is applied. Under this tensile stress, the deformation reflecting the object is called strain.

★ Four Stages of the Stress-Strain Curve

As shown in the diagram below: the stress-strain curve is generally divided into four stages: the elastic region, the flow region, the strain hardening, and the necking fracture.

four stage of tensile stress-strain curve

1 Elastic region

Characteristics: When the stress is below σe, the stress is proportional to the strain of the specimen, the stress is removed and the deformation disappears, i.e. the specimen is in the elastic deformation phase. After the load exceeds the value corresponding to point ‘a’, the tensile curve begins to deviate from the straight line.

tensile stress-strain

Important concept: σe is the elastic limit of the material and represents the maximum stress at which the material remains elastically deformed. In the elastic phase there is a special linear ‘oa’ segment in which there is a linear relationship between σ and ε. This is called the proportional phase, also known as the linear elastic phase. Satisfying Hooke’s law.


E is called the elasticity modulus of the material, generally E = 200 GPa for steel.
The proportional limit σp is the maximum value of stress that obeys Hooke’s law between stress and strain.

σ and ε obey Hooke’s law only when the stress F/A <σp.
For σp<σ<σe, Hooke’s law no longer holds in the ‘ab’ section, but it is still an elastic deformation.
As the difference between σp and σe is not significant, no distinction is made in engineering.

2 Flow region

Characteristics: When the stress exceeds σe to a certain value, the linear relationship between stress and strain is broken and the strain increases significantly, while the stress first decreases and then fluctuates minutely, with small sawtooth line segments appearing close to the horizontal line on the curve. If unloaded, the deformation of the specimen is only partially recovered, while retaining a portion of the residual deformation, i.e. plastic deformation. This indicates that the deformation of the material enters the elasto-plastic deformation phase.

tensile stress-strain

Important concept: σs is called the yield strength or yield point of a material and is an important indicator of plasticity. For materials without significant yielding, in engineering, the value of the stress that produces 0.2% residual deformation is specified as its yield limit.

In tensile testing, if the specimen has yielded, i.e. section ‘bc’ in the diagram above, the specimen continues to elongate even though the load no longer increases, and thus a horizontal interval appears in the tensile curve, a phenomenon known as yielding or flow. The yielding phenomenon is caused by the slippage of crystals in the metal. For materials without yielding, engineering regulations state that the stress corresponding to 0.2% plastic deformation is used as the yield strength, recorded as σ0.2.

3 Strain hardening

Characteristics: When the stress exceeds σs, the specimen undergoes significant and uniform plastic deformation, if the strain on the specimen is to increase, the stress value must be increased. This phenomenon of increasing resistance to plastic deformation as plastic deformation increases is known as work hardening or deformation strengthening.

tensile stress-strain

Important concept: The uniform deformation phase of a specimen ends when the stress reaches σb. This maximum stress σb is called the ultimate strength or tensile strength of the material, which indicates the resistance of the material to maximum uniform plastic deformation, i.e. the maximum stress the material can withstand before tensile damage.

4 Necking fracture

Characteristics: After the stress value of σb, the specimen begins to deform unevenly and form a shrinkage neck, the stress drops and finally the specimen fractures when the stress reaches σf.

tensile stress-strain

Important concept: σf is the fracture strength of the material, which represents the ultimate resistance of the material to plasticity. In general, indicators of the plastic properties of a material are elongation and reduction of area.

Elongation: δ= (L1-L)/L * 100%
Reduction of area: ψ= (A-A1)/A * 100%

L1: length of the specimen after pulling off
L: original length of the specimen
A1: minimum cross-sectional area of the specimen at the fracture
A: original cross-sectional area
The larger the value of δ and ψ, the better the plasticity.

★ Tensile Stress-Strain Curves for Materials with Different Properties

In engineering, materials with an elongation of ≥5% after a break are usually referred to as plastic materials, while materials with an elongation of <5% after a break are referred to as brittle materials. In general, plastic materials can be seen to have a distinct yielding phase, while tensile fracture forms necking. In contrast, brittle materials do not see an obvious yield phase during stretching, and no necking occurs at tensile fracture.

stress-strain curve

A plastic material: very small elastic region.
A ductile material: after the elastic region there is a strange section where ‘necking’ occurs-permanent deformation occurs in this plastic region.
A strong material which is not ductile: Steel wires stretch very little, and break suddenly.
A brittle material: This material is also strong because there is little strain for a high stress. The fracture of a brittle material is sudden with little or no plastic deformation. Glass is brittle stress.

Comparison of the mechanical properties of plastic and brittle materials

Plastic Material Brittle Material
Elongation: δ≥5% Elongation: δ<5%
Large plastic deformation before fracture Very little deformation before fracture
Compressive performance and tensile performance are similar Compressive performance much greater than tensile performance
Suitable for forging and cold working Suitable for foundation elements or shells
Note: The plasticity and brittleness of the material can be altered by changes in manufacturing methods and process conditions.


So that’s all there is to the tensile stress-strain curve, if you want to know how to do the tensile strength test, look here.

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