Main Help → All Commands → Defining Materials → Traction Laws
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:
TriangularTraction
(or 12) - a triangular traction law (MPM only).ExponentialTraction
(or 34) - a exponential traction law (MPM only).CubicTraction
(or 14) - a cubic traction law (MPM only).TrilinearTraction
(or 20) - a trilinear traction law (MPM only).MixedModeTraction
(or 33) - the preferred traction law for mixed mode failure (MPM only).CoupledTraction
(or 23) - a triangular traction law (MPM only).PressureTraction
(or 26) - a constant normal stress traction law (MPM only).LinearTraction
(or 13) - a linear elastic traction law (MPM only).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.
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:
kIe
for slope of the elastic portion of the traction law in pressure units/length units for opening displacement.kIIe
for slope of the elastic portion of the traction law in pressure units/length units for sliding displacement.delpkI
for peak position in mode I relative to the critical COD (delIc
) in mode I (range 0 to 1).delpkII
for peak position in mode II relative to the critical COD (delIIc
) in mode II (range 0 to 1).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).
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.
This material tracks two history variabeles:
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:
alphaI
for α coefficient in the mode I law.alphaII
for α coefficient in the mode II law.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.
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:
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.
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:
You must enter properties for both normal opening (mode I) and shear (mode II) traction laws using the following parameters:
kIe
, kIIe
, delpkI
, delpkII
- These properties are defined in the Triangular Traction Law and they mean the same thing for this traction law, but now apply to the initial linear portion.sigmaI2
and sigmaII2
-
The stress at the second break point in the traction traction law (σ2) in pressure units. Yhe first peak stress is given by sigmaI
or sigmaII
delpkI2
and delpkII2
-
The relative displacement at the location of the second break point in the traction law (δ2/δc). Enter in dimensionless units as fraction of delIc
or delIIc
(0 to 1). It must be ≥ delpkI
or delpkII
.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 σ1+σ2 > 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.
This law can be imagined as modeling the sum of two processes. The total toughness or area for this law is
Jtotal=(1/2)(σ1δ2+σ2(δc-δ1))
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)(σ1δ2-σ2δ1). 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.
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)(σ1δ2-σ2δ1)/(δ2-δ1)
Between δ2 and δc, it will increase linearly with slope or
(1/2)σ2δc/(δc-δ2)
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:
modelI
to set which traction law to use for mode I or normal opening. The options are selected by traction law numerical ID and can be 12 (for Triangular Traction Law), 14 (for Cubic Traction Law), 20 (for Trilinear Traction Law), or 34 (for Exponential Traction Law)modelII
is same as modelI
but used to select law for mode II or tangential opening.NewtonsMethod
- enter 0 (or omit) to used default update methods derived for this coheive law or enter 1 to force use of numerical solution. Numerical methods are generally not needed, but might if the initial elastic regime is too short (i.e., very still law).More details on this law are available on OSUPDocs wiki.
This material tracks the following history variabeles:
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.
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:
stress
- Enter constant normal stress (in pressure units). Use negative stress for a pressure loaded crack.function
- alternatively, the stress can be entered as a user-defined function of time (giving stress in pressure units). If a function is used, any entered stress
property is ignored.minCOD
- specify a minimum, normal, crack opening displacement below which no pressure is applied. This parameter can model pressure induced by a substance that needs room to enter the crack; it is ignored if it is negative (default is -1) (in length units).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.
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:
kIe
for slope of the traction law in pressure units/length units for opening displacement.kIIe
for slope of the traction law in pressure units/length units for sliding displacement.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.
The following traction law properties are common to all types of traction laws. These properties are only used in MPM analyses.
JIc
for area under the traction law for opening displacements in energy release units.JIIc
for area under the traction law for sliding displacements in energy release units.nmix
for the exponent used in the mixed-mode failure criterion for some traction laws. Enter nmix<0
to choose infinity, which is a decoupled failure criterion.sigmaI
for peak stress in the the traction law for opening displacements in pressure units.sigmaII
for peak stress in the the traction law for sliding displacements in pressure units.delIc
for maximum opening displacement in the opening traction law in length units.delIIc
for maximum sliding displacement in the sliding traction law in length units.