Elasto-plastic material modelling and validation

OBJECTIVE:

Material model validation from the given stress-strain curve for elasto-plastic material.

THEORY:

A stress-strain curve for the graphite iron casting is given in the problem statement. Two curves with different material structure have been shown in figure 1.1 below. This response should be captured by either MAT_024(Piecewise linear plasticity) or MAT_018(Power law of plasticity) material model according to the problem statement.

Fig 1.1: Stress-strain curve for the graphite iron casting

MAT_018(Power law of plasticity):

In this material model, the elasto-plastic behaviour of the material is defined by a power law of hardening. This is an isotropic material model. The stress-strain relation is defined by the following equation:

`σ_y=k ε^n` (1)

Where,

`σ_y` = Yield strength

`k` = Strength coefficient

`ε` = Total strain

`n` = Hardening coefficient

So, equation 1 is used to predict the stress-strain behaviour of the elasto-plastic material. The curvature of the curve after the yield point is highly dependent on the hardening coefficient (n). This is one of the simple models to describe metal plasticity.

MAT_024(Piecewise linear plasticity):

This material model also models the elasto-plastic behaviour of the material. Here, the material behaviour can be treated as a bi-linear stress-strain curve or can be treated as a multi-linear curve by defining a set of stress-strain points. The multi-linear behaviour can also be treated by defining a load curve in terms of effective stress vs effective plastic strain. The basic input material parameters for this model are mass density, yield stress, Poisson's ratio, Young’s modulus and tangent modulus. Bi-linear stress-strain curve is the simplest form of representation but the multi-linear curve captures the behaviour more accurately. So, the basic concept here is the continuous smooth stress-strain curve is approximated with small linear steps.

0.2% offset yield stress:

Usually for ductile metals the behaviour changes from elastic to plastic gradually so, the yield point is not well defined and cannot be detected exactly. In these cases, the 0.2% offset yield strength is calculated. The offset yield strength is calculated by constructing a parallel line offset by 0.002 strain to the linear stress-strain line. The given data also shows a gradual change in elastic to plastic behaviour rather than straight horizontal line hence selecting exact yield point is difficult here. The extracted stress-strain data is plotted and the yield stress is calculated by 0.2% offset method for given material which is graphite iron casting.

Figure 1.2 depicts the calculated yield strength by an offset line which is parallel to the initial linear part of the stress-strain curve. The straight line (red colour) is offset by 0.002 strain. The strain values are offset by adding 0.002 extra strain and the stress values are calculated by calculating the young’s modulus and multiplying it with the strain values. Few stress-strain points from the elastic region are considered to calculate the young’s modulus as 147.34 GPa. The intersection of this line to the stress-strain curve gives the 0.2% offset yield stress value. This value can be considered as a yield stress value for the analysis.

But the graphite iron casting does not show high ductility and it can be seen from the provided data that the strain values are also low comparatively. So, this method is not used to find the yield stress for this analysis. This method is widely used in the metal manufacturing industry where the material undergoes plastic or large deformation.

Fig 1.2: Stress-strain curve for graphite iron casting with the calculated yield strength by 0.2% offset yield method

PROCEDURE:

The curve number 2 is selected from the figure 1.1 for this analysis and the material model MAT_024 is used for validation. The true stress-strain data is extracted through the data digitizer into excel. The material shows elasto-plastic behaviour with the elastic and plastic region. The extracted stress(ksi) and strain(%) is converted into stress(N/m2) and strain(-).

The linear piecewise material is used for this analysis. The young’s modulus given in the problem statement is 20.9E+06 psi (144.1004 GPa). The calculated yield stress is used and 1% of elastic modulus is considered as the tangent modulus for the card definition. But the yield stress and tangent modulus values are ignored if the LCSS value is greater than 0. Here we have defined a hardening curve which predicts the behaviour of the material in tension.

The stress-strain data is converted to a kg-mm-ms unit system according to the problem statement. Kg-mm-ms unit system is used hereafter for the analysis. This stress-strain data is converted to effective stress-effective plastic strain data. Now the newly calculated data does not have an elastic strain in it. The hardening curve which is defined in the material card should start ideally with yield strength and effective plastic strain (at yield strength) as a first point. So, a point is manually selected just after the elastic region and considered as the yield point. This selected point is taken as the starting point of the hardening curve (4.53e-3, 2.03e-1). Figure 1.3 shows the defined hardening curve by LCSS option through MAT_024 material card. Figure 1.4 shows the material card assignment with other required properties.

Fig 1.3: Hardening curve defined for the specimen

Fig 1.4: Material properties assigned for the given specimen

Shell section properties have been assigned to the dog-bone specimen with the thickness as 1mm. No contact definition is required for this problem as the displacement and fixed boundary conditions are given directly by selecting nodes.

Displacement is given to the end nodes by *BOUNDARY_PRESCRIBED_MOTION_SET card. Figure 1.5 shows the selected node-set for assigning displacement. Figure 1.6 shows the load curve defined for giving displacement.

Fig 1.5: Displacement boundary condition given for the selected nodes

Fig 1.6: Load curve used for defining the displacement

The other end of a specimen is fixed by single point constraint (SPC) nodes. Two different node sets are defined figure 1.7, (a) shows the node constrained all directions except in Y direction. They are kept free to move along Y direction because the specimen will elongate in the X direction and simultaneously contract in Y and Z direction due to Poisson's ratio. The change in length along Z direction would be very low in this case. Figure 1.7 (b) shows two end nodes constrained in the only Y direction. This is done because the model is symmetric along the Y direction and these nodes should ideally remain at the same Y coordinate during the tensile test.

(a) Nodes constrained in all directions except Y direction

(b) Two end nodes constrained in the only Y direction

Fig 1.7: SPC nodes for assigning fixed boundary condition

This analysis is a slow process and can be considered as a static-structural problem. So, the implicit solver is used to solve this problem. *CONTROL_IMPLICIT_AUTO and *CONTROL_IMPLICIT_GENERAL cards are used with appropriate values. The initial time step value used for analysis is 0.001. The optimum equilibrium iteration is defined 100 with a window of 20 iterations per time step. The end time is defined as 1 by *CONTROL_TERMINATION card. Database cards such as extent binary, binary d3plot and ASCII options are used to get desired results in post-processing.

RESULT AND DISCUSSION:

Effective stress

Figure 1.8 shows the effective stress contours while the specimen is in tension. The middle portion is stressed and the highest effective stress value reaches to 0.2602 Kg/mm ms2. Figure 1.9 depicts the effective stress behaviour with respect to the simulation time of an element in the middle portion of the specimen. We can see the stress remains almost constant after some time and this signifies there is no strain hardening after some time. There is no necking till the end of the analysis so the stress does not dropdown. The stress value has gone above the yield stress value so there would be a plastic deformation in the model.

Fig 1.8: Effective stress in the specimen

Fig 1.9: Effective stress with respect to time

Effective strain:

As discussed above, the stress values are higher than the yield stress so there must be a plastic strain in the model. Figure 1.10 shows the effective plastic strain contour in the specimen. The maximum plastic strain can be seen in the middle portion, this is the portion where necking starts if the specimen is further pulled apart. Figure 1.11 depicts the effective plastic strain versus time behaviour. The plastic strain goes on increasing uniformly until the end of the simulation. The maximum strain value calculated is 0.04157 from figure 1.10.

Fig 1.10: Effective plastic strain contour

Fig 1.11: Effective plastic strain with respect to time

Stress-strain validation:

Figure 1.12 shows the stress-strain behaviour of the dog-bone specimen under tension. The stress almost remains constant after some initial strain hardening. The specimen is stressed till 3.5% of strain. The given stress-strain curve in the problem statement is for low strain values. So, the simulated stress-strain data is considered only till about 0.6% strain. The obtained data from the simulation is true stress and true strain and the given data is also the true stress-strain data therefore, no conversion is needed and both the results can be directly compared.

Figure 1.13 depicts the comparison of the given and the simulated stress-strain data. It can be seen that the elastic part is considered by a linear stress-strain relation. The linear elastic part is calculated directly through the given young’s modulus, this is a one-step calculation. The yielding starts after the first defined point in the hardening curve and stress develop according to the hardening curve further. No hardening is considered after the last defined point in the hardening curve therefore we see a constant stress value after 0.55% - 0.6% strain.

Fig 1.12: Obtained effective stress vs strain plot for the tensile test

Fig 1.13: Validated stress-strain curve using MAT_24 material card

CONCLUSION:

The given stress-strain data is validated with the MAT_024(Piecewise linear plasticity) material model. The stress-strain behaviour after yielding is considered by a hardening curve which is given as an input through MAT_024 card. The MAT_024 material model captures the actual stress-strain behaviour in a similar manner and almost matched the given stress-strain curve.