Connector Functions for Coupled Behavior

The 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

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:

• a norm ($g_N$-type)

  $$T_i\left(\mathbf{c}\right)=s_i\sqrt{\stackrel{N_c}{\sum _{j=1}}{\left({\alpha }_j{c}_j\right)}^2}, \mathrm{or}$$

• a direct sum ($g_S$-type)

  $$T_i\left(\mathbf{c}\right)=s_i\stackrel{N_c}{\sum _{j=1}}{\alpha }_j{c}_j, \mathrm{or}$$

• a Macauley sum ($g_M$-type)

  $$T_i\left(\mathbf{c}\right)=s_i\stackrel{N_c}{\sum _{j=1}}<{\alpha }_j{c}_j>,$$

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 ( $\langle X\rangle =0\mathit{ }\mathrm{if}\mathit{ }X\le 0\mathit{ }\mathrm{and}=X\mathit{ }\mathrm{if}\mathit{ }X>0$ ). 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.

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.

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.

By 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.

Alternatively, you can choose to compute a derived component term as the direct sum of the intrinsic component contributions.

Alternatively, you can choose to compute a derived component term as the Macauley sum of the intrinsic component contributions.

You can specify whether the sign of a derived component term should be positive or negative.

The 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.

When a derived component is used to construct the yield function for a plasticity or friction definition, the following simple requirements must be satisfied:

• All $N_T$ terms of a derived component must be of a compatible type (see Functional Form of the Derived Component); norm-type terms ($g_N$-type) cannot be mixed with direct sum-type terms ($g_S$-type) in the same derived component definition but can be mixed with Macauley sum-type terms ($g_M$-type).

• If all $N_T$ terms are norm-type terms, the sign of each term must be positive (the default).

If $N_T$ is greater than 1, the associated functions (potentials) in which the derived component is used may become non-smooth. More precisely, the normal to the hyper-surface defined by the potential may experience sudden changes in direction at certain locations. In these cases, Abaqus will automatically smooth-out 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.

The spot weld shown in Figure 4 is subjected to loading in the F-direction.

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.

Spot weld connection: derived component definitions.

Therefore, you want to derive the normal and shear components of the force, for example, as follows:

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 (3-direction). If the average cross-section area of the spot weld is A and the beam's second moment of inertia about one of the in-plane 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 cross-section 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 $2r_n$.

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

The 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:

a quadratic form

$$\mathrm{P}\left(\mathbf{c}\right)={\left(\stackrel{N_p}{\sum _{i=1}}{s}_i{P}_i{\left(\mathbf{c}\right)}^2\right)}^{\frac{1}{2}},$$

a general elliptical form

$$\mathrm{P}\left(\mathbf{c}\right)={\left(\sum _{i=1}^{N_p}{s}_i{P}_i{\left(\mathbf{c}\right)}^{{\alpha }_i}\right)}^{\frac{1}{\beta }}, \mathrm{or}$$

a maximum form

$$\mathrm{P}\left(\mathbf{c}\right)=\stackrel{N_p}{\underset{i=1}{max}}{s}_i{P}_i\left(\mathbf{c}\right),$$

where $\mathbf{c}$ is a generic name for the connector intrinsic component values (such as forces, $\mathbf{f}$, or motions, $\mathbf{u}$), $P_i$ 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

where

$H_i\left(X\right)$

is the function used to generate the contribution:

• absolute value (default, $\left|X\right|$),

• Macauley bracket ( $\langle X\rangle =0\mathit{ }\mathrm{if}\mathit{ }X\le 0\mathit{ }\mathrm{and}=X\mathit{ }\mathrm{if}\mathit{ }X>0$ ), or

• identity (X);

$E_i$

is the value of the identified component (intrinsic or derived);

$a_i$

is a shift factor (default 0.0); and

$R_i$

is a scaling factor (default 1.0).

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.

To 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 $\beta$.

Alternatively, you can define the potential as a maximum form.

The 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:

• A norm-type contribution ($P_N$) defined using the absolute value or the Macauley bracket functions or using a combination of norm-type $g_N$ and Macauley sum-type $g_M$ derived components (see Requirements for Constructing a Derived Component Used in Plasticity or Friction Definitions) with any of the available functions.

• A sum-type contribution ($P_S$) defined using an intrinsic component of relative motion or a derived component of type $g_S$ (see Requirements for Constructing a Derived Component Used in Plasticity or Friction Definitions) together with the identity function.

When used in the context of connector plasticity or connector friction, the potential must be constructed such that the following requirements are satisfied:

• All $N_p$ contributions to the potential must be of the same type. Mixed $P_N$ and $P_S$ contributions are not allowed in the same potential definition.

• If all $N_T$ terms are $P_N$-type terms, the sign of each term must be positive (the default).

• The positive numbers $\beta$ and ${\alpha }_i$ cannot be smaller than 1.0 and must be equal (the default).

Referring 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