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What Do Soils and Adhesives Have In Common?

January 15, 2016 By: Michael Bak

At first glance, soils and adhesives are very different materials with no obvious similarities. However, from a structural mechanics point of view, soils and adhesives both can exhibit plastic deformation where hydrostatic stress can have an effect on yielding, and thus their structural responses can be described using Extended Drucker-Prager material models.

Many engineering materials, particularly engineering metals, are assumed to behave linear and elastic up until the shear strain energy in the material reaches a critical value. Once this value, called the yield stress, is reached, the material becomes plastic and exhibits permanent deformations. This definition is the von Mises yield criterion, and the main assumption is that hydrostatic stress, which is the average of the three normal stress components of any stress tensor, and is related to volume change, has no effect on the material yield.

Soil mechanics assumes that the yielding of sand, clay and silt materials is effected by hydrostatic stress. The Drucker-Prager material model is a popular example of a constitutive law developed to describe pressure-dependent von Mises yielding. Plotted in principal stress space, the yield surface is a cone, with the point of the cone in the tension zone, as shown in Figure 1.  Note that as the compressive hydrostatic pressure increases, the yield stress increases, i.e. the cone gets wider.

Figure 1: Classic Drucker-Prager Yield Surface in Principal Space

The classic linear Drucker-Prager yield criterion has the following form:

Most classic Drucker-Prager models, since they were developed for soil mechanics, require input of the cohesion value of the soil, and the angle of internal friction, which are related to the material parameter and yield stress, and can be obtained from soil testing. In addition, classic Drucker-Prager behavior assumes no hardening response upon yielding, i.e. the material is assumed to be perfectly-plastic. This assumption results in excessive plastic dilatation at yielding and the inability to describe hysteretic behavior within the failure surface.

The addition of hardening response, typically referred to as Extended Drucker-Prager, is the feature that has been found to provide more accurate predictive capability for adhesive behavior as well as most soils.  For this model, a flow rule and the stress-strain data describing the hardening must also be defined. An associative flow rule indicates that the direction of plastic flow, called the dilation angle, is perpendicular to the yield surface (and thus equal to the angle of internal friction), resulting in significant volumetric expansion.  A non-associative flow rule, where the dilation angle is less than the angle of internal friction, will result in less volumetric expansion.

Extended Drucker-Prager has also been expanded to include various forms of yield criteria beyond the linear form shown above in Figure 1.  For example, a power law form and hyperbolic form have been shown to provide good prediction for some adhesives.

In ANSYS FEA software, an equivalent linear Extended Drucker-Prager law can be obtained from the classic Drucker-Prager parameters using the relationships below.

 


Since the Extended Drucker-Prager has the additional capabilities of adding a flow rule and hardening information, it is now the standard Drucker-Prager model in ANSYS. The classic Drucker-Prager law is not available for the current 180-series element formulations.

As an example of using Extended Drucker-Prager for an adhesive, a lap joint analysis taken from the UK National Physical Laboratory (NPL) Manual for the Calculation of Elastic-Plastic Material Models Parameters was modeled.  The lap joint geometry is shown in Figure 2.  A two-dimensional plane strain analysis was performed.

Figure 2: Lap Joint Geometry

Material parameters are given for an Extended Drucker-Prager model, including stress-strain data to describe the hardening response. The load versus displacement response was predicted using the ANSYS linear Extended Drucker-Prager model with no hardening, and with the given hardening data. Figure 3 shows the comparison of the two results with the published experimental data, with the hardened response showing excellent correlation.

Figure 3: Comparison of Adhesive Analysis and Experiment

If you have used Drucker-Prager material laws for soils or adhesives, I would love to hear about your experiences. Please feel free to add your comment below.