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23.4 Fabric materials

• “Fabric material behavior,” Section 23.4.1

23.4.1 FABRIC MATERIAL BEHAVIOR

Product: Abaqus/Explicit

References

• “Material library: overview,” Section 21.1.1
• “Elastic behavior: overview,” Section 22.1.1
• “VFABRIC,” Section 1.2.5 of the Abaqus User Subroutines Reference Guide
• *FABRIC
• *UNIAXIAL
• *LOADING DATA
• *UNLOADING DATA
• *EXPANSION
• *DENSITY
• *INITIAL CONDITIONS

Overview

The fabric material model:

• is anisotropic and nonlinear;
• is a phenomenological model that captures the mechanical response of a woven fabric made of yarns in the fill and the warp directions;
• is valid for materials that exhibit two “structural” directions that may not be orthogonal to each other with deformation;
• defines the local fabric stresses as a function of change in angle between the fibers (shear strain) and the nominal strains along the yarn directions;
• allows for the computation of local fabric stresses based on test data or through user subroutine VFABRIC, which can be used to define a complex constitutive model; and
• requires that geometric nonlinearity be accounted for during the analysis step (“General and linear perturbation procedures,” Section 6.1.3), since it is intended for finite-strain applications.

The fabric material model defined based on test data:

• assumes that the responses along the fill and the warp directions are independent of each other and that the shear response is independent of the direct response along the yarns;
• can include separate loading and unloading responses;
• can exhibit nonlinear elastic behavior, damaged elastic behavior, or elastic-plastic type behavior with permanent deformation upon complete unloading;
• can deform elastically to large tensile and shear strains; and
• can have properties that depend on temperature and/or other field variables.

Woven fabrics are used in a number of engineering applications across various industries, including such products as automobile airbags; flexible structures like boat sails and parachutes; reinforcement in composites; architectural expressions in building roof structures; protective vests for military, police, and other security circles; and protective layers around the fuselage in planes.

Woven fabrics consist of yarns woven in the fill and the warp directions. The yarn is crimped, or curved, as it is woven up and down over the cross yarns. The nonlinear mechanical behavior of the fabric arises from different sources: the nonlinear response of the individual yarns, the exchange of crimp between the fill and the warp yarns as they are stretched, and the contact and friction between the yarns in cross directions and between the yarns in the same direction. In general, the fabric exhibits a significant stiffness only along the yarn directions under tension. The tensile response in the fill and warp directions may be coupled due to the crimp exchange mentioned above. Under in-plane shear deformation, the fill and warp direction yarns rotate with respect to each other. The resistance increases with shear deformation as lateral contact is formed between the yarns in each direction. The fabrics typically have negligible stiffness in bending and in-plane compression.

The behavior of woven fabrics is modeled phenomenologically in Abaqus/Explicit to capture the nonlinear anisotropic behavior of the fabric. The planar kinematic state of a given fabric is described in terms of the nominal direct strains in the fabric plane along the fill and the warp directions and the angle between the two yarn directions. The material orthogonal basis and the yarn local directions are illustrated in Figure 23.4.1–1 showing the reference and the deformed configurations.

text_image

E₂ N₂ ψ₁₂⁰ = θ₂⁰ - θ₁⁰ θ₂⁰ N₁ θ₁⁰ E₁

(a) Reference configuration

text_image

e₂ n₂ Ψ₁₂ = Ψ₁₂⁰ - γ₁₂ n₁ e₁

(b) Deformed configuration
Figure 23.4.1–1 Fabric kinematics

The engineering nominal shear strain, \gamma _ { 1 2 } . , is defined as the change in angle, \psi _ { 1 2 } , between the two yarn directions going from the reference to the deformed configuration. The nominal strains along the yarn directions n _ { 1 } and n _ { 2 } in the deformed configuration are computed from the respective yarn stretch values, \lambda _ { 1 } and \lambda _ { 2 } . The corresponding nominal stress components T _ { 1 1 } , T _ { 2 2 } , and T _ { 1 2 } are defined as the work conjugate of the above nominal strains. The fabric nominal stress, , is converted by Abaqus to the Cauchy stress, \sigma ; and the subsequent internal forces arising from the fabric deformation are computed.

You can obtain output of the fabric nominal strains, the fabric nominal stresses, and the regular Cauchy stresses. The relationship between the Cauchy stress, , and the nominal stress, , is

J \sigma = \lambda_ {1} T _ {1 1} \mathbf {n _ {1}} \mathbf {n _ {1}} + \lambda_ {2} T _ {2 2} \mathbf {n _ {2}} \mathbf {n _ {2}} + T _ {1 2} \csc (\psi_ {1 2}) (\mathbf {n _ {1}} \mathbf {n _ {2}} + \mathbf {n _ {2}} \mathbf {n _ {1}}) - T _ {1 2} \cot (\psi_ {1 2}) (\mathbf {n _ {1}} \mathbf {n _ {1}} + \mathbf {n _ {2}} \mathbf {n _ {2}}),

where is the volumetric Jacobian.

Either experimental data or a user subroutine, VFABRIC, can be used to characterize the Abaqus/Explicit fabric material model, providing the nominal fabric stress as a function of the nominal fabric strains. The user subroutine allows for building a complex material model taking into account both the fabric structural parameters such as yarn spacing, yarn cross-section shape, etc. and the yarn material properties. The test data–based fabric model makes some simplifying assumptions but allows for nonlinear response including energy loss. The two models are presented below in detail. Both models capture the wrinkling of fabric under compression only in a smeared fashion.

The application of fabric material in a crash simulation is illustrated in “Side curtain airbag impactor test,” Section 3.3.2 of the Abaqus Example Problems Guide.

Test data–based fabric materials

The fabric material model based on test data assumes that the responses along the fill and the warp directions are independent of each other and that the shear response is independent of the direct response along the yarn. Hence, each component-wise fabric stress response depends only on the fabric strain in that component. Thus, the overall behavior of the fabric consists of three independent component-wise responses: namely, the direct response along the fill yarn to the nominal strain in the fill yarn, the direct response along the warp yarn to the nominal strain in the warp yarn, and the shear response to the change in angle between the two yarns.

Within each component you must provide test data defining the response of the fabric. To fully define the fabric response, the test data must cover all of the following attributes:

• Within a component, separate test data can be defined for the fabric response in the tensile direction and in the compressive direction.
• Within a deformation direction (tension or compression), both loading and unloading test data can be provided.
• The loading and unloading test data can be classified according to three available behavior types: nonlinear elastic behavior, damaged elastic behavior, or elastic-plastic type behavior with permanent deformation. The behavior type determines how the fabric transitions from its loading response to its unloading response.

When elastic, the test data in a particular component can also be rate dependent. When separate loading and unloading paths are required, the test data for the two deformation directions (tension and compression) must be given separately. Otherwise, the data for both tension and compression may be given in a single table.

Input File Usage: Use the following options to define a fabric material using test data:

\begin{array}{l} ^ {* \text { FABRIC }} \\ * \text { UNIAXIAL,   COMPONENT } = \text { component } \\ \end{array}

*LOADING DATA, DIRECTION=deformation direction,

TYPE=behavior type

data lines to define loading data

*UNLOADING DATA

data lines to define unloading data

Repeat all of the options underneath *FABRIC as often as necessary to account for each component and deformation direction.

Specifying uniaxial behavior in a component direction

Independent loading and unloading test data can be provided in each of the three component directions. The components correspond to the response along the fill yarn, the response along the warp yarn, and the shear response.

Input File Usage: Use the following option to define the response along the fill yarn direction:

*UNIAXIAL, COMPONENT=1

Use the following option to define the response along the warp yarn direction:

*UNIAXIAL, COMPONENT=2

Use the following option to define the shear response:

*UNIAXIAL, COMPONENT=SHEAR

Defining the deformation direction

The test data can be defined separately for tension and compression by specifying the deformation direction. If the deformation direction is defined (tension or compression), the tabular values defining tensile or compressive behavior should be specified with positive values of the stress and strain in the specified component and the loading data must start at the origin. If the behavior is not defined in a loading direction, the stress response will be zero in that direction (the fabric has no resistance in that direction).

If the deformation direction is not defined, the data apply to both tension and compression. However, the behavior is then considered to be nonlinear elastic and no unloading response can be specified. The test data will be considered to be symmetric about the origin if either tensile or compressive data are omitted.

Input File Usage: Use the following option to define tensile behavior:

*LOADING DATA, DIRECTION=TENSION

Use the following option to define compressive behavior:

*LOADING DATA, DIRECTION=COMPRESSION

Use the following option to define both tensile and compressive behavior in a single table:

*LOADING DATA

Compressive behavior

In general, a fabric material does not have significant stiffness under compression. To prevent the collapse of wrinkled elements under compressive loading, the specified stress-strain curve should have nonzero compressive stiffness, particularly at larger compressive strains.

Defining loading/unloading component-wise response for a fabric material

To define the loading response, you specify the fabric stress as nonlinear functions of the fabric strain. This function can also depend on temperature and field variables. See “Input syntax rules,” Section 1.2.1, for further information about defining data as functions of temperature and field variables.

The unloading response can be defined in the following different ways:

• You can specify several unloading curves that express the fabric stress as nonlinear functions of the fabric strain; Abaqus interpolates these curves to create an unloading curve that passes through the point of unloading in an analysis.
• You can specify an energy dissipation factor (and a permanent deformation factor for models with permanent deformation), from which Abaqus calculates a quadratic unloading function.
• You can specify an energy dissipation factor (and a permanent deformation factor for models with permanent deformation), from which Abaqus calculates an exponential unloading function.
• You can specify the fabric stress as a nonlinear function of the fabric strain, as well as a transition slope; the fabric unloads along the specified transition slope until it intersects the specified unloading function, at which point it unloads according to the function. (This unloading definition is referred to as combined unloading.)
• You can specify the fabric stress as a nonlinear function of the fabric strain; Abaqus shifts the specified unloading function along the strain axis so that it passes through the point of unloading in an analysis.

The behavior type that is specified for the fabric dictates the type of unloading you can define, as summarized in Table 23.4.1–1. The different behavior types, as well as the associated loading and unloading curves, are discussed in more detail in the sections that follow.

Defining nonlinear elastic behavior

The elastic behavior can be nonlinear and, optionally, rate dependent. When the loading response is rate dependent, a separate unloading curve can also be specified. However, the unloading response need not be rate dependent.

Defining rate-independent elasticity

When the loading response is rate independent, the unloading response is also rate independent and occurs along the same user-specified loading curve as illustrated in Figure 23.4.1–2. An unloading curve does not need to be specified.

Input File Usage: *LOADING DATA, TYPE=ELASTIC

Table 23.4.1–1 Available unloading definitions for the fabric behavior types.

Material behavior type	Unloading definition
	Interpolated	Quadratic	Exponential	Combined	Shifted
Nonlinear elastic (rate-dependent only)	√
Damaged elastic	√	√	√	√
Permanent deformation	√	√	√		√

line

ε	σ
0	0
1	1
2	2
3	3
4	4
5	5
6	6
7	7
8	8
9	9
10	10

Figure 23.4.1–2 Nonlinear elastic rate-independent loading.

Defining rate-dependent elasticity

When the elastic response is rate dependent, both the loading and the unloading curves can be specified. The unloading data can be rate independent or rate dependent and must be specified in a single table; that is, only a single unloading data definition is allowed. If the unloading data are not specified, the unloading occurs along the loading curve specified with the smallest rate of deformation.

Unphysical jumps in the stress due to sudden changes in the rate of deformation are prevented using a technique based on viscoplastic regularization. This technique also helps model relaxation effects in a very simplistic manner, with the relaxation time given as \tau = \mu _ { 0 } + \mu _ { 1 } | \lambda - 1 | ^ { \alpha } , where \mu _ { 0 } , \mu _ { 1 } , and are material parameters and is the stretch. \mu _ { 0 } is a linear viscosity parameter that controls the relaxation time when \lambda \approx 1 . Small values of this parameter should be used; a suggested value is 0.0001s. \mu _ { 1 }

is a nonlinear viscosity parameter that controls the relaxation time at higher values of . The smaller this value, the shorter the relaxation time. The suggested value for this parameter is 0.005s. controls the sensitivity of the relaxation speed to the fabric strain component. Figure 23.4.1–3 illustrates the loading/unloading behavior as the component is loaded at a rate \dot { \varepsilon } _ { 2 } ^ { l } and then unloaded at a rate \dot { \varepsilon } _ { 2 } ^ { u } .

line

ε	σ (ε₃)	σ (ε₃ˡ)	σ (ε₃)	σ (ε₃)	σ (ε₃)	σ (ε₃)	σ (ε₃)	σ (ε₃)	σ (ε₃)	σ (ε₃)	σ (ε₃)
0	0	0	0	0	0	0	0	0	0	0	0
1	~0.5	~0.6	~0.7	~0.8	~0.9	~1.0	~1.1	~1.2	~1.3	~1.4	~1.5
2	~1.0	~1.2	~1.4	~1.6	~1.8	~2.0	~2.2	~2.4	~2.6	~2.8	~3.0
3	~1.5	~1.8	~2.0	~2.2	~2.4	~2.6	~2.8	~3.0	~3.2	~3.4	~3.6
4	~2.0	~2.4	~2.6	~2.8	~3.0	~3.2	~3.4	~3.6	~3.8	~4.0	~4.2
5	~2.5	~3.0	~3.2	~3.4	~3.6	~3.8	~4.0	~4.2	~4.4	~4.6	~4.8
6	~3.0	~3.6	~3.8	~4.0	~4.2	~4.4	~4.6	~4.8	~5.0	~5.2	~5.4
7	~3.5	~4.2	~4.4	~4.6	~4.8	~5.0	~5.2	~5.4	~5.6	~5.8	~6.0
8	~4.0	~4.8	~5.0	~5.2	~5.4	~5.6	~5.8	~6.0	~6.2	~6.4	~6.6
9	~4.5	~5.4	~5.6	~5.8	~6.0	~6.2	~6.4	~6.6	~6.8	~7.0	~7.2
10	~5.0	~6.0	~6.2	~6.4	~6.6	~6.8	~7.0	~7.2	~7.4	~7.6	~7.8

Figure 23.4.1–3 Rate-dependent loading/unloading.

The unloading path is determined by interpolating the specified unloading curves. The unloading need not be rate dependent, even though the loading response is rate dependent. When the unloading is rate dependent, the unloading path at any given component strain and strain rate is determined by interpolating the specified unloading curves.

Input File Usage:

Use the following options when the unloading is also rate dependent:

*LOADING DATA, TYPE=ELASTIC, RATE DEPENDENT,DIRECTION
*UNLOADING DATA, DEFINITION=INTERPOLATED CURVE, RATE DEPENDENT
Use the following options when the unloading is rate independent:
*LOADING DATA, TYPE=ELASTIC, RATE DEPENDENT,DIRECTION
*UNLOADING DATA, DEFINITION=INTERPOLATED CURVE

Defining models with damage

The damage models dissipate energy upon unloading, and there is no permanent deformation upon complete unloading. You can specify the onset of damage by defining the strain above which the material response in unloading does not retrace the loading curve.

The unloading behavior controls the amount of energy dissipated by damage mechanisms and can be specified in one of the following ways:

• an analytical unloading curve (exponential/quadratic);
• an unloading curve interpolated from multiple user-specified unloading curves; or
• unloading along a transition unloading curve (constant slope specified by user) to the user-specified unloading curve (combined unloading).

Input File Usage: Use the following options to define damage with quadratic unloading behavior:

*LOADING DATA, TYPE=DAMAGE, DIRECTION*UNLOADING DATA, DEFINITION=QUADRATIC

Use the following options to define damage with exponential unloading behavior:

*LOADING DATA, TYPE=DAMAGE, DIRECTION*UNLOADING DATA, DEFINITION=EXPONENTIAL

Use the following options to define damage with an interpolated unloading curve:

*LOADING DATA, TYPE=DAMAGE, DIRECTION*UNLOADING DATA, DEFINITION=INTERPOLATED CURVE

Use the following options to specify damage with combined unloading behavior:

*LOADING DATA, TYPE=DAMAGE, DIRECTION *UNLOADING DATA, DEFINITION=COMBINED

Defining onset of damage

You can specify the onset of damage by defining the strain above which the material response in unloading does not retrace the loading curve.

Input File Usage: *LOADING DATA, TYPE=DAMAGE, DAMAGE ONSET=value

Specifying exponential/quadratic unloading

The damage model in Figure 23.4.1–4 is based on an analytical unloading curve that is derived from an energy dissipation factor, (fraction of energy that is dissipated at any strain level). As the fabric component is loaded, the stress follows the path given by the loading curve. If the fabric component is unloaded (for example, at point B), the stress follows the unloading curve . Reloading after unloading follows the unloading curve until the loading is such that the strain becomes greater than , after which the loading path follows the loading curve. The arrows shown in Figure 23.4.1–4 illustrate the loading/unloading paths of this model.