Stress-Strain Diagrams | Engineering Materials - Mechanical Engineering PDF Download

Introduction

The stress-strain curve is the graphical representation of the relationship between the stress applied to a specimen and the resulting strain recorded during a tensile test. Each engineering material produces a characteristic curve that reveals mechanical properties such as the Modulus of Elasticity (E), yield behaviour, ultimate strength and ductility. In a tensile test, the specimen is subjected to an increasing tensile load while changes in length are measured by strain-measuring devices; the recorded load and corresponding elongation data are used to plot stress versus strain.

• Stress (engineering) is defined as the applied load divided by the original cross-sectional area: σ = F / A0.
• Engineering strain is the change of length divided by the original length: ε = ΔL / L0.
• The initial linear portion of the curve obeys Hooke's law and its slope equals the Young's modulus, E, where E = σ / ε in the elastic range.
• Plotting stress versus strain from tensile-test data allows determination of important design parameters and comparison of materials.

Stress-Strain Diagram for a Ductile Material (Mild Steel)

For ductile metals such as mild steel, the stress-strain curve shows distinct regions: an initial proportional (linear) region, an elastic region, yielding, strain hardening up to a maximum load, necking and finally fracture. The specimen initially deforms elastically and returns to its original dimensions when the load is removed. Beyond the elastic limit, plastic deformation occurs and permanent elongation remains after unloading.

• Proportional limit: the stress up to which stress is directly proportional to strain and Hooke's law is valid (linear portion).
• Elastic limit: the maximum stress that can be applied without causing permanent set. Up to this stress the material recovers fully on unloading.
• Yield point / Yield strength: the stress at which material begins to deform plastically. In mild steel a distinct yield point is often observed; beyond this point small increases of stress produce large strains.
• Upper and lower yield points: some steels show an upper yield point followed by a drop to a lower yield point and then a region of yield plateau (Lüders bands may form).
• Work hardening (strain hardening): after yielding, continued plastic deformation increases the stress required to produce further plastic strain; this raises the apparent strength until a maximum load is reached.
• Ultimate tensile strength (UTS): the maximum engineering stress found by dividing the maximum load by the original cross-sectional area.
• Necking: after the peak load, plastic deformation localises and a neck forms; engineering stress falls because of the decreasing instantaneous area while true stress continues to rise locally until fracture.
• Fracture: final separation of the specimen. Ductile fracture is usually preceded by significant plastic deformation and reduction of area.
• Creep: slow increase of strain with time under a constant stress; more important at elevated temperatures and in long-time loading applications.
• Volume conservation: during plastic (uniform) elongation under tensile loading, the specimen approximately conserves volume so decrease in cross-sectional area accompanies lengthening.

Stress-Strain Diagram for a Brittle Material

Brittle materials (for example glass, ceramics, some hard carbides) do not show an extensive plastic region. The curve is often nearly linear until fracture; yield point is not well defined. For such materials the strength is largely determined by the maximum stress at fracture and the fracture strain is small.

• Offset yield method: when a clear yield point is absent, an offset method is used to define yield strength. A line is drawn parallel to the initial elastic slope but offset by a specified strain (commonly 0.1% or 0.2% engineering strain). The intersection of this line with the stress-strain curve defines the yield strength at that offset (e.g., 0.2% proof stress).
• Ductility and brittleness: materials with less than about 5% elongation are often classified as brittle. Short fracture strain and small area under the curve indicate low ductility and low toughness.

Stress-Strain diagram for Cemented tungsten carbide:

Stress-Strain diagram for Plaster of Paris:

Stress-Strain diagram for Soft rubber:

Engineering and True Stress-Strain Diagrams

Engineering (nominal) stress and strain are calculated using the original cross-sectional area and original length respectively: σeng = F / A0 and εeng = ΔL / L0.

True stress and true strain use the instantaneous cross-sectional area and incremental changes in length respectively. True stress is defined as σtrue = F / Ainst and true (logarithmic) strain is εtrue = ln(L / L0) for uniform, continuous deformation.

• Relation during uniform deformation: σtrue = σeng × (1 + εeng) and εtrue = ln(1 + εeng). These relations hold approximately before onset of necking when deformation is uniform along the gauge length.
• Behaviour at necking: after necking begins, instantaneous area decreases markedly only in the neck region; true stress (based on instantaneous area) usually continues to increase up to fracture while engineering stress falls from the UTS onward.
• Use in design: engineering values are commonly used for standard material specifications and design because they are simple and reproducible; true stress-strain data are used for advanced material modelling and where large plastic strains occur.

Some Material Properties Revealed by Stress-Strain Behaviour

• Brittleness - the tendency of a material to fracture without appreciable plastic deformation. It is the opposite of ductility. Examples include glass, concrete and cast iron. Materials with less than ≈5% elongation at fracture are often termed brittle.
• Toughness - the ability of a material to absorb energy up to fracture. It is quantified as the area under the stress-strain curve (energy per unit volume). Tough materials can twist, bend and stretch significantly before failure. Toughness typically decreases with rising temperature for many metals. Toughness is important where parts are subjected to shock or impact loading.

• Stiffness - the resistance of a material to elastic deformation under load. High stiffness means small elastic deformation under large load and corresponds to a high Young's modulus.
• Resilience - the ability of a material to store elastic energy and recover on unloading. The modulus of resilience is the energy absorbed per unit volume up to the elastic limit and is equal to 1/2 × σy × εy for linear elastic behaviour.
• Endurance (fatigue) behaviour - the capacity of a material to withstand varying (cyclic) stresses. The endurance limit is the maximum stress amplitude that can be applied for an effectively infinite number of cycles without causing fatigue failure for some materials. Endurance data are essential for components in reciprocating machines and vibrating structures.
• Anelastic behaviour - time-dependent recoverable deformation. Under constant stress, some recoverable deformation continues as a function of time because of internal relaxation processes. On unloading, part of the deformation recovers over time rather than instantly.
• Viscoelastic behaviour - materials that show both elastic (recoverable) and viscous (permanent, time-dependent) deformation components. Commonly exhibited by non-crystalline organic polymers. Permanent time-dependent deformation in these materials is analogous to creep in crystalline solids.

Key Formulas and Practical Notes

• Engineering stress: σeng = F / A0.
• Engineering strain: εeng = ΔL / L0.
• Young's modulus: E = σ / ε in the elastic region.
• True stress (uniform region): σtrue = F / Ainst = σeng × (1 + εeng).
• True (logarithmic) strain: εtrue = ln(1 + εeng).
• Modulus of resilience (linear elastic): Ur = 1/2 × (σy × εy) = σy² / (2E).
• Toughness: energy absorbed per unit volume = area under the full stress-strain curve up to fracture.

Applications and Design Relevance

• Selection of materials for structural members, springs and machine parts depends on yield strength, UTS, ductility and toughness as read from stress-strain curves.
• Fatigue-sensitive components require materials with adequate endurance limits and good toughness to resist crack initiation and propagation.
• High-temperature components require knowledge of creep behaviour and time-dependent strain at service stresses.
• Engineering specifications commonly quote 0.2% offset yield (proof stress) for materials without a distinct yield point.

Summary

Stress-strain diagrams are fundamental tools that summarise mechanical behaviour under tensile loading. From the same test data one obtains elastic modulus, yield behaviour, strength, ductility, toughness and insight into time-dependent phenomena such as creep and viscoelasticity. Understanding the differences between engineering and true stress-strain representations and recognising the characteristic shapes for ductile, brittle and polymeric materials is essential for material selection and design.