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plates spot weld Fn Fn f3 m1 m2 m3 f1 Fs f2

Figure 31.2.4–5 Spot weld connection: derived component definitions. Therefore, you want to derive the normal and shear components of the force, for example, as follows: $$ F _ {n} = g _ {n} (\mathbf {f}) = < f _ {3} > + (1 / r _ {n}) \sqrt {m _ {1} ^ {2} + m _ {2} ^ {2}}, $$ $$ F _ {s} = g _ {s} (\mathbf {f}) = (1 / r _ {s}) | m _ {3} | + \sqrt {f _ {1} ^ {2} + f _ {2} ^ {2}}. $$ 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 (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 $I _ { 1 1 } ( \mathrm { o r } I _ { 2 2 } ) , r _ { n }$ can be interpreted as the square root of the ratio $I _ { 1 1 } / A \left( \mathrm { o r } I _ { 2 2 } / A \right)$ . 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 $2 r _ { n }$ . 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 $$ I _ {x x} = 3 0. 6 8 $$ $$ A = 1 9. 6 3 $$ $$ r _ {n} = \mathbf {s q r t} \left(I _ {x x} / A\right) $$ $$ r _ {s} = 2. 0 * r _ {n} $$ $$ o r _ {n} = \left(1 / r _ {n}\right) $$ $$ o r _ {s} = \left(1 / r _ {s}\right) $$ \*CONNECTOR DERIVED COMPONENT, NAME=normal, OPERATOR=MACAULEY SUM3 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, $g _ { n } ( \mathbf { f } ) ~ = ~ T _ { 1 } ( \mathbf { f } ) + T _ { 2 } ( \mathbf { f } )$ . The first connector derived component defines the first term $T _ { 1 } ~ = < ~ f _ { 3 } ~ >$ , while the second defines the second term $T _ { 2 } = ( 1 / r _ { n } ) \sqrt { m _ { 1 } { } ^ { 2 } + m _ { 2 } { } ^ { 2 } }$ . Similarly, to define $F _ { s }$ , you would specify the following two connector derived component definitions for the component shear: ```csv * CONNECTOR DERIVED COMPONENT, NAME=shear 6 < or_s > * CONNECTOR DERIVED COMPONENT, NAME=shear 1, 2 1.0, 1.0 ``` # Defining connector potentials Connector potentials are user-defined 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: • friction, • plasticity, or • damage. Consider the case of a SLIDE-PLANE connection in which frictional sliding occurs in the connection plane, as shown in Figure 31.2.4–6. The function governing the stick-slip frictional behavior (see “Connector friction behavior,” Section 31.2.5) can be written as $$ \phi_ {f r i c} (\mathbf {f}) = \mathrm{P} (\mathbf {f}) - \mu F _ {N}, $$ where $\mathrm { P } ( \mathbf { f } )$ is the connector potential defining the pseudo-yield function (the magnitude of the frictional tangential tractions in the connector in a direction tangent to the connection plane on which contact occurs), $\mathrm { F _ { N } }$ is the friction-producing normal (contact) force, and $\mu$ is the friction coefficient. Frictional stick occurs if $\phi _ { f r i c } < 0$ , and sliding occurs if $\phi _ { f r i c } = 0$ . In this case the potential can be defined as the magnitude of the frictional tangential tractions, $$ \mathrm{P} (\mathbf {f}) = \sqrt {f _ {2} ^ {2} + f _ {3} ^ {2}}. $$ 

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normal direction 1 fₙ f₃ 3 2 f₂ sliding with friction in this plane

Figure 31.2.4–6 Friction in the SLIDE-PLANE connection. Connector potentials can also be useful in defining connector damage with a force-based coupled damage initiation criterion. For example, in a connection type with six available components of relative motion you could define a potential $$ \mathrm{P} (\mathbf {f}) = \sqrt {\left(\frac {f _ {1}}{\alpha_ {1}}\right) ^ {2} + \left(\frac {f _ {2}}{\alpha_ {2}}\right) ^ {2} + \left(\frac {f _ {3}}{\alpha_ {3}}\right) ^ {2} + \left(\frac {m _ {1}}{\beta_ {1}}\right) ^ {2} + \left(\frac {m _ {2}}{\beta_ {2}}\right) ^ {2} + \left(\frac {m _ {3}}{\beta_ {3}}\right) ^ {2}}. $$ Damage (with failure) can be initiated when the value of the potential is greater than a user-specified limiting value (usually 1.0). The units of the and coefficients must be consistent with the units of the final product. For example, if the intended units of are newtons, the 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 $$ \phi_ {p l a s} (\mathbf {f}) = \mathrm{P} (\mathbf {f}) - F ^ {0}, $$ where $\mathrm { P } ( \mathbf { f } )$ is the connector potential defining the yield function and $F ^ { 0 }$ is the yield force/moment. The potential could be defined as $$ \mathrm{P} (\mathbf {f}) = \left[ \left(\frac {m a x (F _ {n} , 0)}{R _ {n}}\right) ^ {\beta} + \left(\frac {| F _ {s} |}{R _ {s}}\right) ^ {\beta} \right] ^ {1 / \beta}, $$ 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: \*CONNECTOR POTENTIAL Abaqus/CAE Usage: Use the following input to define connector potentials for friction behavior: Interaction module: connector section editor: Add→Friction: Friction model: User-defined, Slip direction: Compute using force potential, Force Potential Use the following input to define connector potentials for plasticity behavior: Interaction module: connector section editor: Add→Plasticity: Coupling: Coupled, Force Potential Use the following input to define connector potentials for damage behavior: Interaction module: connector section editor: Add→Damage: Coupling: Coupled, Initiation Potential or Evolution Potential # Functional form of the potential The functional form of the potential 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 $$ \mathrm{P} (\mathbf {c}) = \left(\sum_ {i = 1} ^ {N _ {p}} s _ {i} P _ {i} (\mathbf {c}) ^ {2}\right) ^ {\frac {1}{2}}, $$ a general elliptical form $$ \mathrm{P} (\mathbf {c}) = \left(\sum_ {i = 1} ^ {N _ {p}} s _ {i} P _ {i} (\mathbf {c}) ^ {\alpha_ {i}}\right) ^ {\frac {1}{\beta}}, \quad \text {or} $$ a maximum form $$ \mathrm{P} (\mathbf {c}) = \max _ {i = 1} ^ {N _ {p}} s _ {i} P _ {i} (\mathbf {c}), $$ where is a generic name for the connector intrinsic component values (such as forces, , or motions, ), $P _ { i }$ is the $i ^ { \mathrm { t h } }$ contribution to the potential, $N _ { p }$ is the number of contributions, and $\alpha _ { i }$ are positive numbers (defaults $\beta = 2 . 0 , \alpha _ { i } = \beta )$ , and $s _ { i }$ is the overall sign of the contribution $( 1 . 0 - \mathrm { d e f a u l t } , \mathrm { o r } - 1 . 0 )$ . The appropriate component values for 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 . $P _ { i }$ is defined as $$ P _ {i} (\mathbf {c}) = H _ {i} \left(X _ {i} (\mathbf {c})\right) \mathrm{(nosumoni)}, $$ $$ X _ {i} (\mathbf {c}) = \frac {E _ {i} (\mathbf {c}) - a _ {i}}{R _ {i}}, $$ where $H _ { i } ( X )$ is the function used to generate the contribution: • absolute value (default, ), • Macauley bracket $( < X > = 0$ if $X \le 0$ and = X if $X > 0 )$ , 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 $H _ { i } ( X )$ 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 form 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 $\beta$ . 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, $R _ { i } , \ , H _ { i } ( X ) , a _ { i } , s _ { i }$ Use the following option to define a general elliptical form of the potential: \*CONNECTOR POTENTIAL, OPERATOR=SUM, EXPONENT= component name or number, $R _ { i } , \alpha _ { i } , H _ { i } ( X ) , a _ { i } , s _ { i }$ Each data line defines one contribution to the potential, $P _ { i }$ . The function $H _ { i } ( X )$ 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 form Alternatively, you can define the potential as a maximum form.

Input File Usage:

*CONNECTOR POTENTIAL, OPERATOR=MAXcomponent name or number, $R_{i}$ , , $H_{i}(X)$ , $a_{i}$ , $s_{i}$ ...Each data line defines one contribution to the potential, $P_{i}$ . The function $H_{i}(X)$ 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 definitions The connector potential, $\mathrm { P } ( \mathbf { c } )$ , 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 $P _ { N } { \mathrm { - } } \mathrm { t y p e }$ 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). # Example: spot weld Referring to the spot weld shown in Figure 31.2.4–5 and the yield function $\phi _ { p l a s } ( \mathbf { F } )$ 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 # Output The Abaqus/Explicit output variables available for connectors are listed in “Abaqus/Explicit output variable identifiers,” Section 4.2.2. The following variables (available only in Abaqus/Explicit ) are of particular interest when defining connector functions for coupled behavior: CDERF Connector derived force/moment with the connector derived component name appended to the output variable. If the connector derived component is used with connector plasticity, connector friction, and connector damage initiation (type force), the derived components used to form the potential represent forces and this quantity is available for both field and history output. If connector friction is used with contact force, the derived components are not used to form a potential, and the derived force is in fact the connector normal force CNF (which is available for connector history output.) CDERU Connector derived displacement/rotation with the connector derived component name appended to the output variable. If the connector derived component is used with motion type for the connector damage initiation and connector damage evolution, the derived components to form the potential represent displacements and this quantity is available for both field and history output. # 31.2.5 CONNECTOR FRICTION BEHAVIOR Products: Abaqus/Standard Abaqus/Explicit Abaqus/CAE # References • “Connectors: overview,” Section 31.1.1 • “Connector behavior,” Section 31.2.1 • “Connector functions for coupled behavior,” Section 31.2.4 • \*CHANGE FRICTION • \*CONNECTOR BEHAVIOR • \*CONNECTOR DERIVED COMPONENT • \*CONNECTOR FRICTION • \*CONNECTOR POTENTIAL • \*FRICTION • “Defining friction,” Section 15.17.3 of the Abaqus/CAE User’s Guide, in the HTML version of this guide # Overview Frictional effects can be defined in any connector with available components of relative motion. A typical connector might have several pieces that are in relative motion and are contacting with friction. Therefore, both frictional forces and frictional moments may develop in the connector available components of relative motion. To define connector friction in Abaqus, you must specify the following: • the friction law as governed by a friction coefficient; • the contributions to the friction-generating connector contact forces or moments; and • the local “tangent” direction in which the friction forces/moments act. The friction coefficient can be • expressed in a general form in terms of slip rate, contact force, temperature, and field variables; • defined by a static and kinetic term with a smooth transition zone defined by an exponential curve; and • limited by a tangential maximum force, $F _ { m a x } { . }$ , which is the maximum value of tangential force that can be carried by the connector before sliding occurs. Abaqus provides two alternatives for specifying the other aspects of friction interactions in connectors: • Predefined friction interactions for which you need to specify a set of parameters that are characteristic of the connection type for which friction is modeled. Abaqus automatically defines the contact force contributions and the local “tangent” directions in which friction occurs. Predefined friction interactions represent common cases and are available for many connection types (see “Connection-type library,” Section 31.1.5). If desired, known internal contact forces (such as from a press-fit assembly) can be defined as well. • User-defined friction interactions for which you define all friction-generating contact force contributions and the local “tangent” directions along which friction occurs. The user-defined friction interactions can be used if predefined friction is not available for the connection type of interest or if the predefined friction interaction does not adequately describe the mechanism being analyzed. Although more complicated to utilize, user-defined interactions: – are very general in nature due to flexibility in defining arbitrary sliding directions via connector potentials and contact forces via connector derived components; allow for the specification of sliding directions, contact forces, and additional internal contact forces as functions of connector relative position or motion, temperature, and field variables (the internal contact forces can also be dependent on accumulated slip); and – allow for several friction definitions to be used in the same connection applied in different components of relative motion. # Friction formulation in connectors The basic concept of Coulomb friction between two contacting bodies is the relation of the maximum allowable frictional (shear) force across an interface to the contact force between the contacting bodies. In the basic form of the Coulomb friction model, two contacting surfaces can carry shear forces, $\mathrm { F _ { t } }$ , up to a certain magnitude across their interface before they start sliding relative to one another; this state is known as sticking. The Coulomb friction model defines this critical shear force as $\mu \mathrm { F _ { N } }$ , where $\mu$ is the coefficient of friction and $F _ { N }$ is the contact force. The stick/slip calculations determine when a point transitions from sticking to slipping or from slipping to sticking. Mathematically, the relationship can be formalized as $$ \Phi = | \mathbf {F} _ {\mathrm{t}} | - \mu \mathrm{F} _ {\mathrm{N}} \leq 0. $$ Frictional stick occurs if $\Phi < 0 ;$ and sliding occurs if $\Phi = 0$ , in which case the friction force is $\mu \mathrm { F _ { N } }$ Friction in connectors is based on the analogy that contacting surfaces of various parts inside a connector device transmit tangential as well as normal forces across their interfaces. The normal (contact) forces, $\mathrm { F _ { N } }$ , are typically generated by kinematic constraints or by elastic forces/moments in the connector. Connector friction can be used to model tangential (shear) forces, $\mathrm { F _ { t } } .$ in the space spanned by the available components of relative motion for both stick and slip conditions. Figure 31.2.5–1 illustrates the simplest frictional mechanism in connectors, a SLOT connector in a two-dimensional analysis. The local tangent direction in which frictional sliding occurs is the 1-direction (tangential traction $\mathrm { F _ { t } } = f _ { 1 } )$ , and the normal force is developed by the kinematic constraint enforcing the SLOT constraint in the 2-direction, $\mathrm { F _ { N } } = f _ { 2 }$ . The friction model is defined in this case by $$ \Phi = | \mathrm{F} _ {\mathrm{t}} | - \mu \mathrm{F} _ {\mathrm{N}} \leq 0, $$