Modified Crushable Foam

LS-DYNA now offers a modified version of *MAT_CRUSHABLE_FOAM in the form of *MAT_MODIFIED_CRUSHABLE_FOAM . In the new material model, the strain rate effects can be included in the foam a *DEFINE_TABLE . To reduce or eliminate the noise in the evaulation of the new yield stress as a function of strain and strain-rate, LS-DYNA offers a modified strain-rate calculation and also a max variation of strain at every cyle. This is depicted in the following image.

Note
When using this model, the effective plastic strain is replaced with the volumetric strain variable in D3PLOT for easy fringe plotting.

(Click for a larger view)

Determining Prony Series Constants for Modeling Creep and Relaxation

Came across this excellent tutorial that talks in detail the steps involved in determining the Prony series constants.

Determining a Prony Series for a Viscoelastic Material From Time Varying
Strain Data

Unloading Behavior in MAT_083

There are three different ways to model unloading behavior when using material model MAT_083. They are graphically depicted below (these figures may appear in the next release of the LS-DYNA keyword and theory manual).

Option 1 – Default (Click image to enlarge)
In the default option (1), HU=0, and the table is positive which then refers to a series of loading curves. Unloading in this case is then to the loading curve that corresponds to the lowest strain-rate. There is also no rate-dependency so unloading will always occur along the curve that corresponds to the lowest strain-rate (quasi-static or nearly zero).

Option 2 (Click image to enlarge)
In this option, HU=0 and a negative id for the table definition (TBID) will indicate LS-DYNA that first two curves corresponding to the first to strain-rate values in the ascending order forms a closed loop definition and unloading will then occur along the curve that corresponds to lowest strain-rate value.

Option 3 (Click image to enlarge)
In this option, no unloading curve is provided by the user but the unloading curve parameters HU and SHAPE is defined which allows LS-DYNA to compute the unloading curve internally much like what is done in the MAT_LOW_DENSITY foam.

Options 2 and 3 allow for strain-rate dependent unloading and this is shown below.
Strain-rate Dependent Unloading in Option 2 and 3 (Click image to enlarge)

Stress Relaxation in Viscoelastic Material Models

LS-DYNA allows several ways to model of stress relaxation often seen in viscoelastic materials when subjected to a sudden constant strain. The methods to model the stress relaxation is briefly discussed here.

1. Curve Input
When using MAT_GENERAL_VISCOELASTIC material model, one can directly input the time log dependence of the relaxation modulus using LCID parameter. The stress relaxation curve can be either from a shear test or a tensile test in which the stress is divided by the initially applied strain. LS-DYNA then internally perfoms a least square fit to determine the prony series coefficients upto N terms. When using the load curve, users can optionally input the initial BETA value (1/time unit) as the BSTART but is not mandatory.

2. Direct Input of Prony Series Coefficients
Optionally, based on the number of terms desired for the Prony series, the relaxation modulus, G, and the exponential coefficient, BETA, for the ith term can be directly input using the Gi and Bi parameters.

Similar techniques can also be used to model relaxation effects for hyperelastic material models which employ a Prony series with upto 6 terms.

Following figure shows the effects of BETA on the transition between the glassy shear modulus (Short Term) of 40Mpa and the rubbery shear modulus (Long Term, Infinite) of 10MPa. You can download the simple ‘C’ program that generates such a curve for demonstration purposes only.

(Click image to enlarge)

Modeling Off-Axis Dependent Yield Stress for Honeycombs using MAT_126

Honeycomb cell structures are popular among weight conscious designers due to their high strength to weight ratio. In the automotive space, aluminum based honeycomb structures are widely used to represent barriers to simulate a controlled energy absorption. Honeycomb structures are highly anisotropic and requires adequate testing to characterize them. This post focusses on the characterization of the off-axis angle dependency of the yield stress. It is well known that honeycomb structures are stronger along the direction of the cell axis and is considerably weaker in the other two orthogonal directions. The transition of the strength from the strong axis to the weak axis is known to be non-linear and can be captured by using LCA in material model MAT_126 . A typical curve describing the average yield stress as a function of the off-axis angle is shown below.

(Click image to enlarge)

Establishing the yield-stress dependence on the off-axis angle requires tests and in some cases obtaining the values can be quite difficult. In such cases, it is possible to conduct these tests numerically using LS-DYNA to obtain the trend of the off-axis relationship in a very cost and time effective way. The numerical tests are conducted by first preparing a honeycomb specimen which is meshed using shell elements that accurately represents the cells in both the dimensions and the thickness. The shells can be modeled using an appropriate material law to represent the actual material with optional failure parameters. The highlight of this process is in the creation of the specimens for the off-axis angles. Off-axis specimens can be created using the PTRIM option in LS-PREPOST which works by trimming any given component based on the surface generated by a series of IGES lines. A sample part using a IGES trim curve is shown below where the shell structure is rotated by the off-axis angle.

(Click image to enlarge)

Below is an image of honeycomb shell structure and the corresponding off-axis angle specimen at angle using the PTRIM option in LS-PREPOST.

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Curve Extrapolation

Curves are used widely to define a XY data that are used by several entities in LS-DYNA. They help us to define either a time-dependent function, used in loads and boundary conditions, and/or a strain/strain-rate dependent function used frequently in constitutive models. Three most salient features of curve representation, that occur internally in LS-DYNA and which is not quite obvious to the user, is the process of ‘digitization’, ‘extrapolation’ and ‘interpolation’. In this post, the internal extrapolation is addressed.

Need for curve extrapolation
All curve inputs to LS-DYNA have a minimum and a maximum independent variable which we can label them as Xmin and Xmax that defines the independent variable range. During the simulation, when the instantaneous value of the independent variable, Xi, goes beyond this range, internal curve extrapolation, using the last available slope, is performed automatically by LS-DYNA as shown in the curve below. This is required as it is always not possible to anticipate the range of the independent values during the course of the simulation time frame and LS-DYNA helps us by doing the extrapolation to get the appropriate dependent values.

(Click image to enlarge)

Extrapolation in Curves used by Constitutive Models
Most constitutive models use the curve input to allow the definition of the the stress/force as a function of the independent variable which could be strain/displacement. In many cases, uses input the values from the test data which may not necessarily provide the data to cover the entire range of the strain or displacement which necessitates the curve extrapolation. There are several situations where extrapolation is used and they are showed in the following figure.

(Click image to enlarge)

In the case of constitutive models used by springs and discrete beams, care must be taken to define a curve such that it passes the origin (0,0) to model the behavior in both tension and compression. If only one of the phase is defined, then LSDYNA performs extrapolation in the undefined phase using the last available slope nearest the origin in the defined phase as shown below.

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(Click image to enlarge)

Extrapolation in Curves used by Loading and Boundary Conditions.
Keywords that define external loads such as *BOUNDARY and *LOAD skip the extrapolation and instead use a value of ‘0′ every time the instantaneous value of the independent variable, Xi, goes beyond the range of Xmin and Xmax as shown in the following figure. This is done to prevent the generation of non-physical forces due to the extrapolation based on the last available slope. It is therefore the user’s responsibility to ensure that last abscissa value is equal to or greater than the termination time of the simulation. This can be easily ensured by using a *PARAMETRIC_EXPRESSION in which the ENDTIM parameter in CONTROL_TERMINATION is first defined as a parameter variable and then using this as the last available abscissa point in all *DEFINE_CURVE keywords that are referenced by load type keywords. It is also possible to ensure a certain tolerance using the expression < &endtim+0.1*endtim> to extend the last available abscissa beyond the termination time which is necessary to avoid the movement of the nodes back to the initial coordinates when using the *BOUNDARY_PRESCRIBED_{option} keyword.

(Click image to enlarge)

Exceptions to Curve Extrapolation
There is one important exception to the curve extrapolation when defining strain-rate dependent material model using DEFINE_TABLE keyword. To eliminate non-physical scaling of the yield surface, the yield surfaces that correspond to the lowest and highest strain-rates act as the bounds of the yield surface scaling such that the stresses do not exceed these bounds. This is shown in the following figure.

(Click image to enlarge)