Simulation results are evaluated by comparing component stresses and strains to material stress strain curves to understand the component's behavior based on the material's yield or failure limits. **In this blogpost, the entire process from testing through material model selection to curve fitting will be described.**

Content:

Introduction

Material stress-strain curve

What are the details of a stress-strain curve?

Simplified stress-strain curves

Advanced material models

What influences the outcome of the tests?

Convert curves to input variables

## Introduction

Arising stresses and strains heavily depend on the stiffness of the component being simulated. In order to run a structural simulation one must find the properties of the material contributing to stiffness such as elastic modulus or if necessary entire stress-strain curves to understand the material's non-linear behavior. These material properties are obtained with physical testing and the resulting curves are transformed into simulation inputs with the help of material models by curve fitting techniques. The entire process is commonly known as characterization.

## Material stress-strain curve

By now, you hopefully understand that when displacements are applied to nodes in simulations, those are converted into strains and stresses. The outcome of the tests are similar: a small material piece is pulled apart resulting in a stress-strain curve. This curve has a linear and non-linear part. Inputting the single elastic modulus (linear) or the entire curve (non-linear) into the simulation will correlate the applied displacement (strain) to the occuring stresses. One might ask why do we need an entire curve? Well, its not needed as long as the material behaves linearly (small strains) and values are below the yield point. However, this is not always the case but during large strains the curve increases/decreases over a wide strain range and the simulation must account for these varying stress values.

Â

On the following pictures a simple stress-strain curve can be seen, highlighting the difference between linear and non-linear use cases.

If the simulation incorporates only the elastic modulus, the result is valid within the elastic limit. However, if strain surpasses the threshold, stresses will continue increasing linearly based on the single elastic modulus provided (see left picture). This becomes problematic, as beyond the yield limit, the curve flattens (see right picture) indicating loss of stiffness which is not accounted for by the single value but stiffness is over estimated. Consequently, this method is only applicable with small strains.

For the simulation to be valid at large strains, it must include the entire stress-strain curve, encompassing the stress-strain relationship across a broader range rather than being limited to validity solely within the elastic region.

Read more about stiffness here:

__Introduction to FEM (simulaxengineering.com)__

## Details of a stress-strain curve

__Elastic modulus:__Stiffness is typically quantified by the modulus of elasticity or Young's modulus. In the field of materials science and engineering, the modulus of elasticity is a measure of a material's resistance to deformation and is expressed as the ratio of stress to strain. It represents the slope of the stress-strain curve within the elastic region of the material and is indicated with capital E.

Â

__Yield strength:__Has the value in the simulation gone beyond the yield limit of the material? Is the question popping up amongst engineers daily. This critical point marks the transition from linear to non-linear behavior, triggering plasticity. Plasticity entails irreversible deformation within the model, causing the component to retain its altered shape even after the load is no longer present. For engineers, yielding is undesirable, as exceeding the yield limit in any part of the component results in permanent shape change. Beyond this threshold, the component loses its ability to withstand the applied load effectively, leading to significant deformation.

Â

__Tensile strength:__this is also called the ultimate tensile strength, as this is the stress level where components damage. This might or might not align with the fracture stress depending on the material type but damage definitely takes place.

Â

The use of last 2 heavily depend on the used material as ductile ones will yield but on the contrary brittle ones will fracture. Meaning, that yield strength is mostly relevant in soft materials (metals and soft polymers) meanwhile tensile strength is used in case of brittle ones as fx. Concrete and brittle materials.

## Simplified stress-strain curve

The most accurate results are obtained with greater material details however, material curves are not always available. In cases when a linear model is not enough but the full curve is not available, a stress-strain curve can also be fabricated using material datasheet values. These curves will deviate from the measured material curve but would serve most of the purpose.

Several types of curves can be fabricated depending on the intention:

Â

__Bi-linear stress-strain curve__

It is probably the simplest representation of a measured stress-strain curve. Here, the yield stress and tensile strength values are plotted and connected with a straight line. As the plot also shows, some hardening effects are lost and the results must be evaluated conservative though, its a good starting point.

__Elastic-perfectly plastic stress-strain curve__

This curve is applicable when the sole concern is whether plasticity occurs or not, but the model can lose validity beyond the yield point. It provides a straightforward means of constraining stresses, unlike a linear approach where stresses could escalate significantly. In this scenario, plastic strain values lack validity as the curve is flat, indicating zero stiffness. However, it can still be examined to understand whether stresses have surpassed the yield limit.

Read more about how material stiffness relates to element stiffness here: __Introduction to FEM (__simulaxengineering.com)

## Advanced material model

Hyper elastic (HE) models are used to capture the quasi-static behavior of materials under large deformations, handling strains up to 300%, while describing material stiffness as a function of strain. On the other hand, linear viscoelastic (LVE) models address the dynamic properties of materials, focusing on its dependence on frequency and temperature. Although these models can account for specific conditions, a more comprehensive approach is needed to accurately describe material stiffness across all loading scenarios and temperatures. This requires a model that considers large strains, temperature variations, and frequency effects simultaneously. Such models, which integrate all these factors, are known as non-linear viscoelastic (NLVE) models.

Â

Testing of the followings can happen but not limited to by tensile, stress relaxation or DMA (dynamical-mechanical analysis) tests at several strains, strain-rates, temperatures and frequency range depending on the problem.

__Hyper elasticity__

The hyper elastic material model represents the materialâ€™s nonlinear elasticity, but no time-dependence. Elasticity is a linear phenomenon illustrated by a linearly increasing straight line. In case of hyper elasticity we are talking about the same material curve except that the curve is non-linear yet, still elastic.

__Viscoelasticity__

Viscoelastic materials possess a unique combination of viscous (liquid-like) and elastic (spring-like) properties, allowing them to undergo both types of deformation. This dual behavior makes them time-dependent, meaning their response to stress or strain varies based on the duration and rate of loading. The viscous aspect reflects the materialâ€™s ability to absorb and dissipate energy over time, while the elastic component allows it to store and recover energy. Due to this energy absorption and dissipation capacity, viscoelastic materials are often highly effective at damping vibrations and mitigating dynamic forces, making them ideal for applications where controlling energy flow or reducing mechanical vibrations is crucial. Available and commonly used models:

__HE+LVE__Â is the simplest non-linear visco model and probably the most commonly used.Â

By combining the hyper elastic (non-linear model) with a linear visco approach, many of the polymer's strain-rate; frequency and time dependency can be described. However, this combined model remains limited to the elastic range, as it does not account for plastic deformation. To fully capture the materialâ€™s behavior, including permanent deformation, a more comprehensive non-linear viscoelastic model, such as the Parallel Rheological Framework (PRF), is required.

Â

The __Parallel Rheological Framework (PRF)__ builds onto the HE+LVE and accounts for the non-linear relaxation (creep). This framework is capable of capturing the full spectrum of material responses, including instantaneous elastic recovery, delayed time-dependent viscous recovery, and irreversible plastic deformation too. Furthermore, even in its simplest implementation, the PRF can be effectively calibrated using uniaxial testing data over a range of temperatures and strain rates, making it a powerful tool for predicting material performance under diverse mechanical and thermal conditions.

The ability of non-linear viscoelastic models to be calibrated is largely dependent on the quality and range of available test data. Depending on the nature and timescale of the problem, either simple tensile curves (at several strain rates); stress relaxation tests or dynamic mechanical analysis (DMA) may be suitable for capturing the material's time dependent behavior. However, each method has its own limitations, which engineers must consider when selecting the appropriate approach for model calibration and analysis.

## What influences the outcome of the tests?

Material's behavior differ when tested at __different strain-rates__. Slower test speed will result in a flatter curve compared to a rapid one where the curve increases instantanenously and break. The same polymer can be ductile at low strain-rate but brittle and high rates [see middle picture above]. This behavior applies to both metals and polymers however, for different reasons. Metals possess this behavior due to their molecular structure meanwhile, polymers are viscoelastic.

Â

__Direction of test__ (tension/compression): Metals behave similarly in tension and compression however, polymers are much stronger in compression in general and this is to take into account when building models.

Â

__The size__ and geometry of the test specimen can affect the distribution of stresses and strains. Larger specimens may exhibit different mechanical properties due to internal defects, inhomogeneities, or how stress is distributed across the material. If the wall of the component is 0.2 [mm], a test specimen with 2 [mm] thickness will yield different results.

At higher __temperatures__, polymers become softer and more ductile, while at lower temperatures, they become stiffer and more brittle. The transition between glassy and rubbery states can also be temperature-dependent.

Â

The __frequency__ at which the load is applied (in tests like dynamic mechanical analysis, DMA) influences how the viscoelastic material responds. At higher frequencies, the material behaves stiffer, while at lower frequencies, the material has more time to flow and shows more viscous (damping) behavior.

## Convert curves to input variables

Once the material has been tested properly, the acquired test results are turned into simulation inputs using the selected material model equation.

Â

In engineering, things can be described by mathematical models and It's the same when evaluating material behavior. Mathematical expressions exist to characterize material behavior, comprising equations with __parameters that need to be determined__ by fitting an artificial curve to the measured data. Subsequently, these parameters are input into simulation software.

Â

As an example, the simplest Neo-Hooken hyper elastic material characterization is shown here. Neo-Hooken model is a first order reduced polynomial model. It is also the simplest hyper elastic material model which only contains the uniaxial deformation of an elastomer. The use of this model is limited to 20% and normally advised to choose a higher order fitting such as the Yeoh model. However, due to its simplicity it is a perfect choice to represent the curve fitting technique.

As shown in the first plot, the fitted curve is based on the equation TU, which includes the parameters **C10** and **Î»**. Here, **Î»** represents the normal strain+1, while **C10**â€‹ is the unknown parameter being determined. The value of **C10**â€‹ is found by fitting the curve, minimizing the difference between the experimental data and the model using relative or absolute error calculations after an initial guess. Achieving a perfect fit is unlikely, especially with a first-order polynomial. For a more accurate fit, higher-order equations are needed.

## Summary

One must choose a method that fits the test, the nature of the simulated problem and the used simulation type. Metals can easily be modelled with one of the simple approaches presented above. In case of polymers, engineers need to decide whether viscoelastic models are needed to characterize polymers or a simple elastic-plastic model will do. If viscoelastic approach is chosen then do we choose linear or non-linear approach and is stress relaxation data sufficient or should DMA be used. Once the approach has been chosen, which of the mathematical model can best describe the specific boundary condition?

One can go down the easy road and choose the simplest tests and characterization method with a high safety factor to be on the safe side but will that provide the most optimized design? Certainly not! It will always be a compromise in terms of time, money, precision and computational power.

## Comments