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Cracks with Traction Law Materials

MPM implements explicit cracks by defining a series or massless particles that define the crack path. The method is called the CRAMP algorithm. The CRAMP algorithm takes care of the crack geometry and can handle crack-surface contact or imperfect interface contact. In addition, MPM can implement traction laws on the crack surfaces for cohesive zone modeling by assigning a traction law to one or more crack particles along the crack. The traction laws can be assigned when creating the crack or during crack propagation (i.e., new crack surfaces can be dynamically create traction laws). This section explains the possible traction laws. See the crack creation and crack propagation commands for how to use traction laws on cracks.

A traction law is defined using a Material command. See that command for (matid) and (name) parameters and then use the (type) parameter to define the type of traction law. The current options for (type) (by name or number) are:

The Material command should be followed by lines that contain the traction law properties. Each property should be on a separate line and when the last property is specified, enter a line with the single Done Command. The various traction laws expect different properties. The required properties for each traction law type are given in following sections.

More documentation on traction laws can be found in the OSUPDocs wiki including important information on calculating mode I and mode II energy release rates when modeling crack growth using cohesive zones. It is relatively easy to add more traction laws; contact the developer if you need one.

Triangular Traction Law

The traction laws in pure mode I or pure mode II each have the following triangular shape:

There are separate traction laws for opening displacement (mode I) and sliding displacement (mode II). Enter the traction law properties using the following properties:

You must specify exactly two (no more or less) of the common properties for each mode (i.e., two of JIc, sigmaI, and delIc and two of JIIc, sigmaII, and delIIc). Whichever property is not specified will be calculated from the other properties using the following relations:

You must enter zero or one of the terms above for each mode (i.e., zero or one of kIe and delpkI and zero or one of kIIe and delpkII). If one is provided, the other will be calculated from k = σ/(δpeak δc). A valid traction law requires δpeak ≤ 1. If neither k nor δpeak are entered, the peak will be located at δpeak = 0.225926299. This location provides the closest match to the cubic traction law for area under the curve as a function of δ. The initial slope will be set to k = σ/(0.225926299 δc).

Mixed-Mode Failure

These entered traction laws give mode I or mode II traction law for pure mode I or pure mode II loading. For mixed-mode loading, the normal and and shear tractions still depend only on normal and shear displacements, respectively, but the conditions for failure change. Under pure mode loading, failure occurs with the relevant displacement reaches the critical displacement. For mixed-mode loading, failure occurs when:

(GI/JIc)nmix + (GII/JIIc)nmix = 1

where GI and GII are areas under mode I and mode II traction laws up to current displacements. Now this failure criterion will mean that failure may occur when normal or shear tractions are nonzero and result in a sudden drop to zero traction. This approach is identical to the one adopted by Thouless et. al to handle mixed-mode loading and appears to work well. The alternative is to used coupled traction laws. The preferred las is the Mixed Mode Traction Law. Another option is the Coupled Traction Law but this model severely limits options for choosing normal and tangential cohesive laws.

Traction History Variables

This material tracks two history variabeles:

  1. Maximum normal opening displacement (or equal to normal δe prior to initiation).
  2. Maximum shear opening displacement (or equal to tangential δe prior to initiation)

Exponential Traction Law Material (Type=34)

This traction law assumes a linear elastic response up to σc followed by a scaled exponential decrease to reach zero at critical crack opening displacement (COD) of δc.

Notice how the shape changes with α. In the limit of α=0, this law is identical to the Triangular Traction Law (and that one should be used instead because it is more efficient). Increasing α causes traction to drop faster, which might simulate increasingly brittle response. There are separate traction laws for opening displacement (mode I) and sliding displacement (mode II). Enter the traction law properties using the following properties:

For mode I, you have to enter alphaI, exactly two of sigmaI, delpkI, and kIe and either JIc or delIc. For mode II, you must enter the corresponding mode II properties. The α values must be greater than zero.

This material tracks two history variables.

More details for this law are provided on OSUPDocs wiki.

Cubic Traction Law

This traction law is a cubic relation where the slope is zero at δc and the peak always occurs at δc/3. This traction law was first proposed by Needleman. It has a convenient smooth shape for numerical calculations and the zero slope at δc may be desirable. There is no physical basis to claim it is more realistic than other traction laws. The function and shape are given below:

σ = (27/4) σmax (δ/δc)(1-(δ/δc))2

You must specify exactly two (no more or less) of the common properties for each mode (i.e., two of JIc, sigmaI, and delIc and two of JIIc, sigmaII, and delIIc). Whichever property is not specified will be calculated from the other properties using the following relations:

The failure criterion under possible mixed-mode conditions is identical to the one used in the triangular traction law.

This material tracks two history variables.

Trilinear Traction Law

This traction law is a piece-wise linear relation with two break points (i.e., three linear pieces) The traction laws in pure mode I or pure mode II each have the following trilinear shapes with arbitrarily-specifiable break points:

triangular traction law

You must enter properties for both normal opening (mode I) and shear (mode II) traction laws using the following parameters:

You must enter exactly 5 of the seven properties for each mode (e.g, for mode I, five of JIc, sigmaI, delIc, kIe, delpkI, sigmaI2, and delpkI2). Furthermore, if one of the 5 is kIe (or kIIe for mode II), then either sigmaI or delpkI (but not both) must be specified. The unspecified parameters will be calculated. The final law must satisfy δ1 ≤ δ2 ≤ δc, δc > 0, σ1 ≥ 0, σ2 ≥ 0, and σ12 > 0.

These entered traction laws give mode I or mode II traction law for pure mode I or pure mode II loading. The failure under mixed loading uses the same criterion explained above.

This material tracks two history variables.

Failure Mechanisms

This law can be imagined as modeling the sum of two processes. The total toughness or area for this law is

Jtotal=(1/2)(σ1δ22c1))

The area under the first peak and bounded by the dotted red line in the above figure, is the total energy released by the first process. It is equal to J1=(1/2)(σ1δ22δ1). The remaining area is associated with the second process. It is equal to J2=(1/2)σ2δc. For example, the first peak might be associated with crack tip fracture toughness while the tail models a process zone such as fiber bridging. In fact, in the limit as σ1 → ∞ and δ2 → 0, a simulation with a pre-existing cohesive zone should approach a simulation propagating a crack tip using fracture mechanics along with a linear softening traction law having σ2 as the cohesive stress. Both these simulations can be run as two numerical methods for solving the same problem.

R Curve

The theoretical J vs δ curve will be flat up to δ1 and then will increase linearly up to δ2 with a slope of

(1/2)(σ1δ22δ1)/(δ21)

Between δ2 and δc, it will increase linearly with slope or

(1/2)σ2δc/(δc2)

Mixed Mode Traction Law Material (Type=34)

This Mixed Mode Traction law model allows completely independent normal and tangential traction laws and all calculations are coupled and remain valid during mixed-mode loading. The normal and tangential laws can be selecting from Triangular Traction Law, Exponential Traction Law, Cubic Traction Law and Trilinear Traction Law. The properties assigned to the laws are totally independent. It can even use different law types for normal and tangential traction.

You must enter properties for both normal opening (mode I) and shear (mode II) traction laws using the following parameters:

More details on this law are available on OSUPDocs wiki.

Traction History Variables

This material tracks the following history variabeles:

  1. The damage variable D
  2. A damage parameter characterizing mode I damage, δn
  3. A damage parameter characterizing mode II damage, δt
  4. Cumulative work energy
  5. Normal crack opening displacement (un)
  6. Tangential crack opening displacement (ut)

Coupled Traction Law Material

This traction law implements effective displacement methods for mixed mode failure form the literature published prior to the recommended approach implemented in the mixed mode traction law. This model is much more limited and only made available for comparisons. It can do either linear ("saw tooth") decay or exponential decay. More details are currently only available on OSUPDocs wiki.

Pressure Traction Law Material

This traction law creates a constant stress that is normal to the crack surface. It models a pressure loaded crack. The stress can be entered in one of two ways:

Experience shows that applying full stress from the start of calculation can cause large, local vibrations and lead to unstable results. If it often better to use the function option and ramp the stress to desired value.

This material tracks no history variabeles.

Linear Elastic Traction Law

This traction law is linear elastic and never fails as pictured below:

There are separate slopes for opening displacement (mode I) and sliding displacement (mode II) entered using the following properties:

If either slope is not entered, it will be set to zero or no tractions. This traction law never fails and therefore never releases energy. This material tracks no history variabeles.

Common Traction Law Properties

The following traction law properties are common to all types of traction laws. These properties are only used in MPM analyses.