ProductsAbaqus/StandardAbaqus/ExplicitAbaqus/CAE Defining derived components for connector elementsThe definition of coupled behavior in connector elements beyond simple linear elasticity or damping often requires the definition of a resultant force involving several intrinsic (1 through 6) components or the definition of a “direction” not aligned with any of the intrinsic components. These userdefined resultants or directions are called derived components. The forces and motions associated with these derived components are functions of the forces and motions in the intrinsic relative components of motion in the connector element. Consider the case of a SLOT connector for which frictional effects (see Connector friction behavior) are defined in the only available component of relative motion (the 1direction). The two constraints enforced by this connection type will produce two reaction forces (${f}_{2}$ and ${f}_{3}$), as shown in Figure 1. Both forces generate friction in the 1direction in a coupled fashion. Figure 1. Resultant contact force in a SLOT
connector.
A reasonable estimate for the resulting contact force is $${F}_{\mathrm{derived}}^{\mathrm{contact}}=g\left(\mathbf{f}\right)=\sqrt{{f}_{2}^{2}+{f}_{3}^{2}},$$ where $\mathbf{f}$ is the collection of connector forces and moments in the intrinsic components. The function $g\left(\mathbf{f}\right)$ can be specified as a derived component. Resultant forces that can be defined as derived components may take more complicated forms. Consider a BUSHING connection type for which a tensile (Mode I) damage mechanism with failure is to be specified in the 1direction. You may wish to include the effects of the axial force ${f}_{1}$ and of the resultant of the “flexural” moments ${m}_{2}$ and ${m}_{3}$ in defining an overall resultant force in the axial direction upon which damage initiation (and failure) can be triggered, as shown in Figure 2. Figure 2. Resultant axial force in a BUSHING
connector.
One choice would be to define the resultant axial force as $${F}_{\mathrm{derived}}^{\mathrm{axial}}=g\left(\mathbf{f}\right)=\left{f}_{1}\right+\alpha \sqrt{\left({m}_{2}^{2}+{m}_{3}^{2}\right)},$$ where $\alpha $ is a geometric factor relating translations to rotations with units of one over length. The function $g\left(\mathbf{f}\right)$ can be specified as a derived component. A derived component can also be interpreted as a userspecified direction that is not aligned with the connector component directions. For example, if the motionbased damage with failure criterion in a CARTESIAN connection with elastic behavior does not align with the intrinsic component directions, the damage criterion can be defined in terms of a derived component representing a different direction, as shown in Figure 3. Figure 3. Userdefined direction in a CARTESIAN
connector.
One possible choice for the direction could be $${u}_{\mathrm{derived}}^{\mathrm{transf}}=g\left(\mathbf{u}\right)={a}_{1}{u}_{1}+{a}_{2}{u}_{2}+{a}_{3}{u}_{3},$$ where $\mathbf{u}$ is the collection of connector relative motions in the components and ${a}_{1}$, ${a}_{2}$, and ${a}_{3}$ can be interpreted as direction cosines ($cos\left({\alpha}_{1}\right)$, $cos\left({\alpha}_{2}\right)$, $cos\left({\alpha}_{3}\right)$). The function $g\left(\mathbf{u}\right)$ can be specified as a derived component. Functional form of the derived componentThe functional form of a derived component $\left(g\right)$ in Abaqus is quite general; you specify its exact form. The derived component is specified as a sum of terms $$g\left(\mathbf{c}\right)=\stackrel{{N}_{T}}{\sum _{i=1}}{T}_{i}\left(\mathbf{c}\right),$$ where $\mathbf{c}$ is a generic name for the connector intrinsic component values (such as forces, $\mathbf{f}$, or motions, $\mathbf{u}$), ${T}_{i}$ is the ${i}^{\mathrm{th}}$ term in the sum, and ${N}_{T}$ is the number of terms. The appropriate component values for $\mathbf{c}$ are selected depending on the context in which the derived component is used. ${T}_{i}$ is also a summation of several contributions and can take one of the following three forms:
where ${s}_{i}$ is the term's sign (plus or minus), ${\alpha}_{j}$ are scaling factors, ${c}_{j}$ is the ${j}^{\mathrm{th}}$ component of $\mathbf{c}$, and $<X>$ is the Macauley bracket ($<X\ge 0\mathit{}\mathrm{if}\mathit{}X\le 0\mathit{}\mathrm{and}=X\mathit{}\mathrm{if}\mathit{}X0$). In general, the units of the scaling factors ${\alpha}_{j}$ depend on the context. In most cases they are either dimensionless, have units of length, or have units of one over length. The scaling factors should be chosen such that all the terms in the resulting derived component have the same units, and these units must be consistent with the use of the derived component later on in a connector potential or connector contact force. Defining a derived component with only one term (N_{T} = 1)Connector derived components are identified by the names given to them. If one term (${T}_{1}$) is sufficient to define the derived component g, specify only one connector derived component definition. Input File Usage CONNECTOR DERIVED COMPONENT, NAME=derived_component_name Abaqus/CAE Usage Connector derived component names are not supported in Abaqus/CAE; you define the individual derived component terms. Use the following input to define a connector derived component term for a frictiongenerating userdefined contact force: Interaction module: connector section editor: AddFriction: Friction model: Userdefined, Contact Force, Specify component: Derived component, click Edit to display the derived component editor: click Add and select components Use the following input to define a connector derived component term as an intermediate result in a connector potential function: Interaction module: connector section editor: AddFriction, Plasticity, or Damage: potential contribution editors: Specify derived component, click Edit to display the derived component editor: click Add and select components Defining a derived component containing multiple terms (N_{T} > 1)If several terms (${T}_{1}$, ${T}_{2}$, etc.) are needed in the overall definition of the derived component g, you must define the individual terms. Input File Usage You must specify ${N}_{T}$ connector derived component definitions with the same name to define the individual terms. All definitions with the same name will be summed together to produce the desired derived component g. See the spot weld example below for an illustration. CONNECTOR DERIVED COMPONENT, NAME=derived_component_name CONNECTOR DERIVED COMPONENT, NAME=derived_component_name ... Abaqus/CAE Usage Connector derived component names are not supported in Abaqus/CAE; you define the individual derived component terms. Interaction module: derived component editor: click Add and select components. Repeat, adding terms as necessary. Specifying a term in the derived component as a normBy default, a derived component term is computed as the square root of the sum of the squares of each intrinsic component contribution. If the term has only one contribution (${N}_{C}=1$), the norm has the same meaning as the absolute value. Input File Usage CONNECTOR DERIVED COMPONENT, NAME=derived_component_name, OPERATOR=NORM (default) For example, the following input can be used to define the ${F}_{\mathrm{derived}}^{\mathrm{axial}}$ component discussed above: CONNECTOR DERIVED COMPONENT, NAME=axial 1 1.0, ** ${T}_{1}=\sqrt{{\left(1.0*{f}_{1}\right)}^{2}}=\left1.0*{f}_{1}\right$CONNECTOR DERIVED COMPONENT, NAME=axial 5, 6 $\alpha $, $\alpha $ ** ${T}_{2}=\sqrt{{\left(\alpha {m}_{2}\right)}^{2}+{\left(\alpha {m}_{3}\right)}^{2}}$ The axial derived component is ${F}_{\mathrm{derived}}^{\mathrm{axial}}={T}_{1}+{T}_{2}$. Abaqus/CAE Usage Interaction module: derived component editor: Add: Term operator: Square root of sum of squares Specifying a term in the derived component as a direct sumAlternatively, you can choose to compute a derived component term as the direct sum of the intrinsic component contributions. Input File Usage CONNECTOR DERIVED COMPONENT, NAME=derived_component_name, OPERATOR=SUM For example, the following input can be used to define the ${u}_{\mathrm{derived}}^{\mathrm{transf}}$ component discussed above: CONNECTOR DERIVED COMPONENT, NAME=transf, OPERATOR=SUM 1, 2, 3 ${a}_{1}$, ${a}_{2}$, ${a}_{3}$ ** ${T}_{1}={a}_{1}{u}_{1}+{a}_{2}{u}_{2}+{a}_{3}{u}_{3}$ The transf derived component is ${u}_{\mathrm{derived}}^{\mathrm{transf}}={T}_{1}$. Abaqus/CAE Usage Interaction module: derived component editor: Add: Term operator: Direct sum Specifying a term in the derived component as a Macauley sumAlternatively, you can choose to compute a derived component term as the Macauley sum of the intrinsic component contributions. Input File Usage CONNECTOR DERIVED COMPONENT, NAME=derived_component_name, OPERATOR=MACAULEY SUM For example, the following input can be used to define the first term of the normal component of the force (${F}_{n}$) in the spotweld example discussed below: CONNECTOR DERIVED COMPONENT, NAME=normal, OPERATOR=MACAULEY SUM 3 1.0 ** ${T}_{1}=<{f}_{3}>$ Abaqus/CAE Usage Interaction module: derived component editor: Add: Term operator: Macauley sum Specifying the sign of a termYou can specify whether the sign of a derived component term should be positive or negative. Input File Usage Use one of the following options: CONNECTOR DERIVED COMPONENT, NAME=derived_component_name, SIGN=POSITIVE (default) CONNECTOR DERIVED COMPONENT, NAME=derived_component_name, SIGN=NEGATIVE Abaqus/CAE Usage Interaction module: derived component editor: Add: Overall term sign: Positive or Negative Defining the derived component contributions to depend on local directionsThe scaling factors ${\alpha}_{j}$ used in the definition of the derived component can depend on the relative positions or constitutive displacements/rotations in several component directions, as described in Defining nonlinear connector behavior properties to depend on relative positions or constitutive displacements/rotations. See the first example in Connector friction behavior for an illustration. Input File Usage Use the following option to define a connector derived component that depends on components of relative position: CONNECTOR DERIVED COMPONENT, INDEPENDENT COMPONENTS=POSITION Use the following option to define a connector derived component that depends on components of constitutive displacements or rotations: CONNECTOR DERIVED COMPONENT, INDEPENDENT COMPONENTS=CONSTITUTIVE MOTION Abaqus/CAE Usage Interaction module: derived component editor: Add: Use local directions: Independent position components or Independent constitutive motion components Requirements for constructing a derived component used in plasticity or friction definitionsWhen a derived component is used to construct the yield function for a plasticity or friction definition, the following simple requirements must be satisfied:
If ${N}_{T}$ is greater than 1, the associated functions (potentials) in which the derived component is used may become nonsmooth. More precisely, the normal to the hypersurface defined by the potential may experience sudden changes in direction at certain locations. In these cases, Abaqus will automatically smoothout the defined functions by slightly changing the derived component functional definition. These changes should be transparent to the user as the results of the analysis will change only by a small margin. Example: spot weldThe spot weld shown in Figure 4 is subjected to loading in the Fdirection. Figure 4. Loading of a spot weld connection.
The connector chosen to model the spot weld has six available components of relative motion: three translations (components 1–3) and three rotations (components 4–6). You have chosen this connection type because you are modeling a general deformation state. However, you would like to define inelastic behavior in the connection in terms of a normal and a shear force, as shown in Figure 5, since experimental data are available in this format. Figure 5. Spot weld connection: derived component definitions.
Therefore, you want to derive the normal and shear components of the force, for example, as follows: $${F}_{n}={g}_{n}\left(\mathbf{f}\right)=<{f}_{3}>+\left(1/{r}_{n}\right)\sqrt{m_{1}{}^{2}+m_{2}{}^{2}},$$ $${F}_{s}={g}_{s}\left(\mathbf{f}\right)=\left(1/{r}_{s}\right)\left{m}_{3}\right+\sqrt{f_{1}{}^{2}+f_{2}{}^{2}}.$$ In these equations ${r}_{n}$ and ${r}_{s}$ have units of length; their interpretation is relatively straightforward if you consider the spot weld as a short beam with the axis along the spot weld axis (3direction). If the average crosssection area of the spot weld is A and the beam's second moment of inertia about one of the inplane axes is ${I}_{11}$ (or ${I}_{22}$), ${r}_{n}$ can be interpreted as the square root of the ratio ${I}_{11}/A$ (or ${I}_{22}/A$). Furthermore, if the crosssection is considered to be circular, ${r}_{n}$ becomes equal to a fraction of the spot weld radius. In all cases ${r}_{s}$ can be taken to be $2{r}_{n}$. The reasoning above for the interpretation of the calibration constants in the equations is only a suggestion. In general, any combination of constants that would lead to good comparisons with other results (experimental, analytical, etc.) is equally valuable. To define ${F}_{n}$, you would specify the following two connector derived component definitions, each with the same name: PARAMETER ${I}_{xx}$=30.68 A=19.63 ${r}_{n}$=sqrt(${I}_{xx}/A$) ${r}_{s}$=$2.0*{r}_{n}$ $o{r}_{n}=\left(1/{r}_{n}\right)$ $o{r}_{s}=\left(1/{r}_{s}\right)$ CONNECTOR DERIVED COMPONENT, NAME=normal, OPERATOR=MACAULEY SUM 3 1.0 CONNECTOR DERIVED COMPONENT, NAME=normal 4, 5 $<o{r}_{n}>$, $<o{r}_{n}>$ The $<>$ symbols denote that $o{r}_{n}$ is specified using a parameter definition. The normal force derived component ${F}_{n}$ is defined as the sum of two terms, ${g}_{n}\left(\mathbf{f}\right)={T}_{1}\left(\mathbf{f}\right)+{T}_{2}\left(\mathbf{f}\right)$. The first connector derived component defines the first term ${T}_{1}=<{f}_{3}>$, while the second defines the second term ${T}_{2}=\left(1/{r}_{n}\right)\sqrt{m_{1}{}^{2}+m_{2}{}^{2}}$. Similarly, to define ${F}_{s}$, you would specify the following two connector derived component definitions for the component shear: CONNECTOR DERIVED COMPONENT, NAME=shear 6 $<o{r}_{s}>$ CONNECTOR DERIVED COMPONENT, NAME=shear 1, 2 1.0, 1.0 Defining connector potentialsConnector potentials are userdefined mathematical functions that represent yield surfaces, limiting surfaces, or magnitude measures in the space spanned by the components of relative motion in the connector. The functions can be quadratic, general elliptical, or maximum norms. The connector potential does not define a connector behavior by itself; instead, it is used to define the following coupled connector behaviors:
Consider the case of a SLIDEPLANE connection in which frictional sliding occurs in the connection plane, as shown in Figure 6. Figure 6. Friction in the SLIDEPLANE
connection.
The function governing the stickslip frictional behavior (see Connector friction behavior) can be written as $${\varphi}_{fric}\left(\mathbf{f}\right)=\mathrm{P}\left(\mathbf{f}\right)\mu {F}_{N},$$ where $\mathrm{P}\left(\mathbf{f}\right)$ is the connector potential defining the pseudoyield function (the magnitude of the frictional tangential tractions in the connector in a direction tangent to the connection plane on which contact occurs), ${\mathrm{F}}_{\mathrm{N}}$ is the frictionproducing normal (contact) force, and $\mu $ is the friction coefficient. Frictional stick occurs if ${\varphi}_{fric}<0$, and sliding occurs if ${\varphi}_{fric}=0$. In this case the potential can be defined as the magnitude of the frictional tangential tractions, $$\mathrm{P}\left(\mathbf{f}\right)=\sqrt{{f}_{2}^{2}+{f}_{3}^{2}}.$$ Connector potentials can also be useful in defining connector damage with a forcebased coupled damage initiation criterion. For example, in a connection type with six available components of relative motion you could define a potential $$\mathrm{P}\left(\mathbf{f}\right)=\sqrt{{\left(\frac{{f}_{1}}{{\alpha}_{1}}\right)}^{2}+{\left(\frac{{f}_{2}}{{\alpha}_{2}}\right)}^{2}+{\left(\frac{{f}_{3}}{{\alpha}_{3}}\right)}^{2}+{\left(\frac{{m}_{1}}{{\beta}_{1}}\right)}^{2}+{\left(\frac{{m}_{2}}{{\beta}_{2}}\right)}^{2}+{\left(\frac{{m}_{3}}{{\beta}_{3}}\right)}^{2}}.$$ Damage (with failure) can be initiated when the value of the potential $\mathrm{P}$ is greater than a userspecified limiting value (usually 1.0). The units of the $\alpha $ and $\beta $ coefficients must be consistent with the units of the final product. For example, if the intended units of $\mathrm{P}\left(\mathbf{f}\right)$ are newtons, the $\alpha $ coefficients are dimensionless while the $\beta $ coefficients have units of length. Connector potentials can take more complicated forms. Assume that coupled plasticity is to be defined in a spot weld, in which case a plastic yield criterion can be defined as $${\varphi}_{plas}\left(\mathbf{f}\right)=\mathrm{P}\left(\mathbf{f}\right){F}^{0},$$ where $\mathrm{P}\left(\mathbf{f}\right)$ is the connector potential defining the yield function and ${F}^{0}$ is the yield force/moment. The potential could be defined as $$\mathrm{P}\left(\mathbf{f}\right)={\left[{\left(\frac{max\left({F}_{n},0\right)}{{R}_{n}}\right)}^{\beta}+{\left(\frac{\left{F}_{s}\right}{{R}_{s}}\right)}^{\beta}\right]}^{1/\beta},$$ where ${F}_{n}$ and ${F}_{s}$ could be the named derived components normal and shear defined in the example in Defining derived components for connector elements above. If ${F}^{0}$ has units of force and ${F}_{n}$ and ${F}_{s}$ also have units of force, ${R}_{n}$ and ${R}_{s}$ are dimensionless. Input File Usage Abaqus/CAE Usage Use the following input to define connector potentials for friction behavior: Interaction module: connector section editor: AddFriction: Friction model: Userdefined, Slip direction: Compute using force potential, Force Potential Use the following input to define connector potentials for plasticity behavior: Interaction module: connector section editor: AddPlasticity: Coupling: Coupled, Force Potential Use the following input to define connector potentials for damage behavior: Interaction module: connector section editor: AddDamage: Coupling: Coupled, Initiation Potential or Evolution Potential Functional form of the potentialThe functional form of the potential $\mathrm{P}$ in Abaqus is quite general; you specify its exact form. The potential is specified as one of the following direct functions of several contributions:
where $\mathbf{c}$ is a generic name for the connector intrinsic component values (such as forces, $\mathbf{f}$, or motions, $\mathbf{u}$), ${P}_{i}$ is the ${i}^{\mathrm{th}}$ contribution to the potential, ${N}_{p}$ is the number of contributions, $\beta $ and ${\alpha}_{i}$ are positive numbers (defaults $\beta =$ 2.0, ${\alpha}_{i}=\beta $), and ${s}_{i}$ is the overall sign of the contribution (1.0 – default, or −1.0). The appropriate component values for $\mathbf{c}$ are selected depending on the context in which the potential is used in. The positive exponents (${\alpha}_{i}$, $\beta $) and the sign ${s}_{i}$ should be chosen such that the contribution ${P}_{i}$ yields a real number. ${P}_{i}$ is a direct function of either an intrinsic connector component (1 through 6) or a derived connector component. Since derived components are ultimately a function of intrinsic components (see Defining derived components for connector elements), the contribution ${P}_{i}$ is ultimately a function of $\mathbf{c}$. ${P}_{i}$ is defined as $${P}_{i}\left(\mathbf{c}\right)={H}_{i}\left({X}_{i}\left(\mathbf{c}\right)\right)\left(\mathrm{no}\mathrm{sum}\mathrm{on}\mathrm{i}\right),$$ $${X}_{i}\left(\mathbf{c}\right)=\frac{{E}_{i}\left(\mathbf{c}\right){a}_{i}}{{R}_{i}},$$ where
The function ${H}_{i}\left(X\right)$ can be the identity function only if ${\alpha}_{i}=\beta =1.0$. The units of the various coefficients in the equations above depend on the context in which the potential is used. In most cases the coefficients in the equations above are either dimensionless, have units of length, or have units of one over length. In all cases you must be careful in defining potentials for which the units are consistent. Defining the potential as a quadratic or general elliptical formTo define a general elliptical form of the potential, you must specify the inverse of the overall exponent, $\beta $. You can also define the exponents ${\alpha}_{i}$ if they are different from the default value, which is the specified value of $\beta $. Input File Usage To define a quadratic form of the potential, you can omit specifying $\beta $ since its default value is 2.0. Use the following option: CONNECTOR POTENTIAL component name or number, ${R}_{i}$, , ${H}_{i}\left(X\right)$, ${a}_{i}$, ${s}_{i}$ ... Use the following option to define a general elliptical form of the potential: CONNECTOR POTENTIAL, OPERATOR=SUM, EXPONENT=$\beta $ component name or number, ${R}_{i}$, ${\alpha}_{i}$, ${H}_{i}\left(X\right)$, ${a}_{i}$, ${s}_{i}$ ... Each data line defines one contribution to the potential, ${P}_{i}$. The function ${H}_{i}\left(X\right)$ can be ABS (absolute value and the default), MACAULEY (Macauley bracket), or NONE (identity). Abaqus/CAE Usage Interaction module: connector section editor: friction, plasticity, or damage behavior option: Force Potential, Initiation Potential, or Evolution Potential: Operator: Sum, Exponent: 2 (for quadratic form) or $\beta $ (for elliptical form), select Add and enter data for the potential contribution. Repeat, adding contributions as necessary. Defining the potential as a maximum formAlternatively, you can define the potential as a maximum form. Input File Usage CONNECTOR POTENTIAL, OPERATOR=MAX component name or number, ${R}_{i}$, , ${H}_{i}\left(X\right)$, ${a}_{i}$, ${s}_{i}$ ... Each data line defines one contribution to the potential, ${P}_{i}$. The function ${H}_{i}\left(X\right)$ can be ABS (absolute value and the default), MACAULEY (Macauley bracket), or NONE (identity). Abaqus/CAE Usage Interaction module: connector section editor: friction, plasticity, or damage behavior option: Force Potential, Initiation Potential, or Evolution Potential: Operator: Maximum, select Add and enter data for the potential contribution. Repeat, adding contributions as necessary. Requirements for constructing a potential used in plasticity or friction definitionsThe connector potential, $\mathrm{P}\left(\mathbf{c}\right)$, can be defined using intrinsic components of relative motion, derived components, or both. A particular contribution to the potential may be one of the following two types:
When used in the context of connector plasticity or connector friction, the potential must be constructed such that the following requirements are satisfied:
Example: spot weldReferring to the spot weld shown in Figure 5 and the yield function ${\varphi}_{plas}\left(\mathbf{F}\right)$ defined above, you would define this potential using the derived components normal and shear with the following input: PARAMETER ${R}_{n}$=0.02 ${R}_{s}$=0.05 $\beta $=1.5 CONNECTOR POTENTIAL, EXPONENT=$\beta $ normal, ${R}_{n}$, , MACAULEY shear, ${R}_{s}$, , ABS OutputThe Abaqus/Explicit output variables available for connectors are listed in Abaqus/Explicit output variable identifiers. The following variables (available only in Abaqus/Explicit ) are of particular interest when defining connector functions for coupled behavior:
