Modeling Inelasticity With SimScale: Plasticity and Creep
The simplest form of analysis in solid mechanics is linear static analysis. Here the deformations in the material are assumed to be small and Hooke’s law (or a linear relation between stress and strain) can be assumed. However, as the load increases, the material behavior (or otherwise stress-strain relationship) becomes nonlinear demonstrating effects like plasticity and creep.
There are several types of nonlinearities like material, geometric, and boundary nonlinearities. Thin structures undergo large rotations and displacements in spite of the material satisfying Hooke’s law and these are known as geometric nonlinearities. Alternatively, when aspects of contact phenomenon are involved, nonlinearity exists in the boundary conditions (How to Set Up Boundary Conditions in your Simulation?). However, the most common type of nonlinearity is the material nonlinearity observed due to the nonlinearity in the stress-strain behavior.
The most challenging problems are of course those involving a combination of the above-mentioned nonlinearities and possibly others. For example: In a car crash simulation, the displacements are large and the stress-strain behavior is nonlinear. Additionally, inertia and contact (with other bodies and self-contact) must be considered during the simulation.
Material nonlinearity: Plasticity
Material nonlinearity is related to aspects of nonlinear elasticity and inelastic phenomenon like plasticity, creep etc. At small stresses, both materials follow a linear stress-strain behavior. Upon an increase in stress, the stress-strain relation is nonlinear but there is no distinction made between loading and unloading except for the sign in nonlinear elastic materials. Upon complete unloading, no residual strains are observed.
In contrast, in elastoplastic materials, as the stress increases beyond a threshold (known as yield limit) the stress-strain behavior becomes nonlinear. Upon unloading, the elastoplastic material leads to a new branch of the stress-strain curve where the material behaves elastic again. However, upon complete unloading, a residual strain (known as plastic strain) remains. This is described as plasticity behavior observed in the material.
Fig 01: Stress-strain curves for nonlinear elastic material (left) and elastoplastic material (right)
In reality, the curved part observed in elastoplastic materials is quite complicated and thus a number of approximations, known as hardening models, are made to describe this region. There are several models that can be applied to large-scale simulations to replicate the plastic effect. The most common hardening rules are isotropic, kinematic and mixed. Over the decades, several complicated hardening rules and plasticity models have evolved. Nevertheless, it is important to understand the physical cause that leads to plasticity and this varies according to the material. For example, while in metals, it is the movement of dislocations; in polymers, it is due to chain rearrangements.
In the presence of hardening, both the elastic and plastic strain continuously increase beyond the yield limit. Or in other words, the strains increase with increasing stresses but at a lower rate than below the yield limit. Alternatively, in the presence of softening effects, the strains continue to increase in spite of a decrease in stress. The difference between hardening and softening is as shown in Fig. 02.
Fig 02: Strain-stress behavior with strain hardening (left) and softening (right)
As shown in Fig. 02, strain hardening is shown through a linear hardening model where strains increase with increasing stresses (though at a lower rate than below the yield point). In contrast, strain softening shows that strains continue to increase in spite of decreasing stresses.
Important Terms in Plasticity
Consider a ductile material being subjected to an uniaxial tensile loading. As shown in Fig. 03, as the load increases from Pt. 1 and up to Pt. 2, the material behaves elastically. In other words, upon unloading, it follows the same stress-strain curve it followed during loading.
Fig 03: Typical stress-strain curve obtained during uniaxial tensile testing of ductile material
However, upon loading beyond Pt. 2, the material no longer behaves elastically. Or otherwise, upon unloading follows a different path. This is shown as unloading from Pt. 4. This point at which the material becomes inelastic is termed as “Yield Point” and the corresponding stress as “Yield Stress”.
At any point, the stress is a symmetric 2nd order tensor or represented through a symmetric matrix. This implies that there are six independent values. These stresses are defined in the global coordinate system. Now, if a different coordinate system is selected, the stresses need to be transformed using a transformation matrix.
If a transformation matrix is defined such that resulting stress matrix is diagonal, then the diagonal elements are the principal stresses. The three columns define the three vectors that form the new coordinate system.
The physical meaning is that the three planes to which these three vectors are normals are only subject to tensile loading (& not shear). The tensile loads on the three planes are given by the three principal stresses.
von Mises Stress
As we already discussed, the stress tensor has six independent values. But as seen above there is only one yield stress value obtained for a material. So the question is how to choose which stress value that needs to be compared with the yield stress to determine if the material has yielded.
The case of uniaxial tensile loading was simple. This can be treated as a one-dimensional problem and only the stress along the axis of loading can be considered for comparison. But this does not hold true in a general case.
There are several criteria, available in literature, that can be used as a comparison with the yield stress. SimScale uses von Mises yield condition. The von Mises stress can be expressed in terms of the stress tensor components as
This von Mises stress is used as a comparison to check if the material has yielded. If the von Mises stress is larger than the yield stress, then the material can be said to have yielded.
Linear Hardening: Plasticity
At present, SimScale offers usage of Isotropic hardening rules to describe plastic effects. Linear hardening means that beyond the yield point, the stress-strain relation is still linear. However, the modulus for loading is different from that of unloading.
As shown in Fig. 04, the slope of the stress-strain curve beyond the yield limit is still positive but less than the original.
Fig 04: Material demonstrating linear hardening beyond the yield point
SimScale allows modeling plastic effect using the linear hardening model. As shown in Fig. 05, the inputs are Young’s modulus, Poisson ratio and von Mises stress (or otherwise, the yield limit). The yield limit can also be provided through the input of experimental stress-strain data. This is explained in more detail in the SimScale documentation.
Fig 05: Input for plastic material in SimScale
Material nonlinearity: Creep
Creep deformation is a time-dependent deformation. When a constant stress (or force) are applied the strain initially reaches a particular level. Further on, if the stress is continued to be applied, the strain gradually increases (in spite of the constant force / stress). Such a behavior is irreversible and more commonly observed in viscoelastic materials like rubbers and polymeric materials. Creep is not necessarily a damaging phenomenon. For example creep in concrete allows for reduction of tensile stresses which could have otherwise led to cracks and fracture.
Fig 06: Typical strain behavior of a viscoelastic material over time
As shown in Fig. 06, there are three stages of creep. The primary stage demonstrates an initiation of the creeping process and is relatively slow. In contrast, the tertiary stage indicates the formation of necking and is considerably rapid. The secondary stage is when the material undergoes deformation in a reasonably stable manner and is well understood. SimScale facilitates for creep deformation in the primary and secondary stages.
As observed above, to simulate creep phenomenon, a relation needs to be defined regarding how the strains change with time and stress applied. Several phenomenon based models have been defined in SimScale.
Fig 07: Options provided by SimScale for Creep behavior modeling
As shown, all the formulations are equally stable. However, choosing the appropriate law depends on the availability of experimental results using which the required parameters can be obtained.
- Norton is the simplest and depends only on the stress. Here the power law is considered with m = 0 or i.e. the dependence of the time is not considered.
- Time hardening uses the power law with dependence on both the stress and time considered as shown Fig. 07.
- Strain hardening is more complicated. Alongside dependence on stresses, an additional dependence of “creep strain” is also considered as
For more information on the additive decomposition of strains into elastic – plastic – creep parts etc are discussed in more detail in SimScale documentation.
Overall, SimScale allows for modeling plastic and creep effects that are commonly observed in metals, polymers, rubber materials etc. These options already provide an excellent first-order approximation for modeling these phenomena in a wide genre of materials.
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