Secant Modulus

In material testing and engineering, understanding how a material responds to applied stress is essential for predicting performance, ensuring safety, and optimising design. The secant modulus is a measure of material stiffness, calculated from the slope of a line drawn between the origin of a stress-strain curve and a specific point on that curve.

Unlike other measures of modulus that apply only within the strictly elastic region of deformation, the secant modulus can be taken at any point along the curve. This makes it especially valuable for materials that do not exhibit a perfectly linear elastic region. It is widely used in structural analysis, civil engineering, and quality assurance to evaluate mechanical properties beyond the initial elastic range.

By quantifying stiffness in this way, the secant modulus bridges the gap between purely theoretical elastic properties and the actual, sometimes non-linear, behaviour of materials under real-world loading conditions.

Different moduli are used to describe the stiffness of materials, each calculated at different points on the stress-strain curve:

• Tangent modulus represents the slope of the curve at a single point. It reflects the instantaneous stiffness of the material at that strain level and is often used when analysing changes during plastic deformation.
• Initial modulus, also known as Young’s modulus or elastic modulus, is the slope within the perfectly elastic region. It assumes a linear relationship between stress and strain.
• Secant modulus accounts for both elastic and inelastic deformation between the origin and a chosen point, producing a single value that represents average stiffness over that range.

While the tangent modulus changes continuously as deformation progresses, the secant modulus provides an averaged measure of stiffness between two defined points. This is particularly useful in design work where loads may cause deformation beyond the elastic region but well before structural failure.

The elastic modulus measures stiffness only within the initial linear elastic range, where deformation is fully recoverable. The secant modulus may be calculated at any point, including regions where permanent deformation has begun.

This distinction is important in practical engineering. For example, reinforced concrete rarely exhibits a perfectly linear stress-strain relationship. In such cases, the secant modulus offers a more realistic representation of stiffness under working loads.

Accurate calculation requires stress-strain data, typically obtained from tensile, compressive, or flexural testing. The process is:

• Conduct the test - Perform a suitable method such as tensile testing for metals or compressive testing for concrete. Record load and deformation.
• Plot the stress-strain curve - Stress is the applied force divided by the original cross-sectional area. Strain is the change in length divided by the original length.
• Select the point of interest - Choose a point according to the design requirement or standard.
• Determine the slope - The secant modulus is the slope from the origin to the selected point.

Formula:

$$E_s = \frac{\sigma}{\epsilon}$$

Where:

$E_s$ = secant modulus (Pa or N/mm²) $\sigma$ = stress at the chosen point (Pa or N/mm²) $\epsilon$ = strain at the chosen point (dimensionless)

Example:

If stress is 150 MPa at a strain of 0.005, then:

$E_s = 150\ \text{MPa}/0.005 = 30,000\ \text{MPa}$

Care is needed when selecting the reference point, as values will change depending on where it is taken. Industry standards often define exact strain levels for consistency.

The secant modulus is valuable where material behaviour beyond the elastic limit is relevant:

• Concrete design - Used for deflection and crack width calculations, as concrete exhibits non-linear behaviour at relatively low stress.
• Polymer components - Accounts for significant non-linear elasticity in polymers, improving performance predictions for components under long-term load.
• Metal fatigue analysis - Helps predict life expectancy and prevent fatigue failure by assessing stiffness beyond the initial elastic range.

For example, in reinforced concrete bridge decks, deflection under live loads is predicted using the secant modulus at a specific working stress. In composite materials, values at different load stages reveal stiffness degradation as fibres fail. Elastomeric suspension bushings are also assessed this way to predict real-world deformation.

Consistent measurement is supported by recognised standards:

• ASTM C469 - Static modulus of elasticity and Poisson’s ratio of concrete in compression. Defines secant modulus between zero and a specified stress.
• ISO 527 - Plastics, determination of tensile properties. Includes secant modulus measurement for plastics and composites.
• ASTM D790 - Flexural properties of unreinforced and reinforced plastics, allowing secant modulus calculation in flexural tests.

Adhering to these ensures reliable and repeatable results compatible with design codes.

Mecmesin testing systems such as the OmniTest and MultiTest-dV, combined with VectorPro software, provide precise load and displacement measurement, enabling accurate secant modulus calculations. VectorPro automates the process from test sequence design to calculation, reducing operator error and providing traceable results.

Real-time graphing within the software gives immediate visual confirmation of stress-strain behaviour, allowing engineers to validate material performance quickly. Systems can be configured to meet ASTM, ISO, and other standards, supporting a wide range of materials from concrete and metals to polymers and composites.

Selecting the right approach to secant modulus testing depends on the material, applicable standards, and the intended use of the results. Speak to an expert at Mecmesin about configuring your OmniTest or MultiTest-dV system for accurate, standards-compliant secant modulus testing tailored to your application.