김경종
327 lines
26 KiB
Markdown
327 lines
26 KiB
Markdown
<!-- source-page: 861 -->
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# 32.5.6 DEFINING THE CONSTITUTIVE RESPONSE OF COHESIVE ELEMENTS USING ATRACTION-SEPARATION DESCRIPTION
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Products: Abaqus/Standard Abaqus/Explicit Abaqus/CAE
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# References
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• “Cohesive elements: overview,” Section 32.5.1
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• “Defining the constitutive response of cohesive elements using a continuum approach,” Section 32.5.5
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• \*COHESIVE SECTION
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• \*DAMAGE EVOLUTION
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• \*DAMAGE INITIATION
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• “Defining damage,” Section 12.9.3 of the Abaqus/CAE User’s Guide, in the HTML version of this guide
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• Chapter 21, “Adhesive joints and bonded interfaces,” of the Abaqus/CAE User’s Guide
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# Overview
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The features described in this section are primarily intended for bonded interfaces where the interface thickness is negligibly small. In such cases it may be straightforward to define the constitutive response of the cohesive layer directly in terms of traction versus separation. If the interface adhesive layer has a finite thickness and macroscopic properties (such as stiffness and strength) of the adhesive material are available, it may be more appropriate to model the response using conventional material models. The former approach is discussed in this section, while the latter approach is discussed in “Defining the constitutive response of cohesive elements using a continuum approach,” Section 32.5.5.
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Cohesive behavior defined directly in terms of a traction-separation law:
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• can be used to model the delamination at interfaces in composites directly in terms of traction versus separation;
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• allows specification of material data such as the fracture energy as a function of the ratio of normal to shear deformation (mode mix) at the interface;
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• assumes a linear elastic traction-separation law prior to damage;
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• can be used in combination with linear viscoelasticity in Abaqus/Explicit (“Defining viscoelastic behavior for traction-separation elasticity in Abaqus/Explicit” in “Time domain viscoelasticity,” Section 22.7.1) to describe rate-dependent delamination behavior;
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• assumes that failure of the elements is characterized by progressive degradation of the material stiffness, which is driven by a damage process;
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• allows multiple damage mechanisms; and
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<!-- source-page: 862 -->
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• can be used with user subroutine UMAT in Abaqus/Standard or VUMAT in Abaqus/Explicit to specify user-defined traction-separation laws.
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# Defining constitutive response in terms of traction-separation laws
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To define the constitutive response of the cohesive element directly in terms of traction versus separation, you choose a traction-separation response when defining the section behavior of the cohesive elements.
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Input File Usage: \*COHESIVE SECTION, RESPONSE=TRACTION SEPARATION
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Abaqus/CAE Usage: Property module: Create Section: select Other as the section Category and Cohesive as the section Type: Response: Traction Separation
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# Linear elastic traction-separation behavior
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The available traction-separation model in Abaqus assumes initially linear elastic behavior (see “Defining elasticity in terms of tractions and separations for cohesive elements” in “Linear elastic behavior,” Section 22.2.1) followed by the initiation and evolution of damage. The elastic behavior is written in terms of an elastic constitutive matrix that relates the nominal stresses to the nominal strains across the interface. The nominal stresses are the force components divided by the original area at each integration point, while the nominal strains are the separations divided by the original thickness at each integration point. The default value of the original constitutive thickness is 1.0 if traction-separation response is specified, which ensures that the nominal strain is equal to the separation (i.e., relative displacements of the top and bottom faces). The constitutive thickness used for traction-separation response is typically different from the geometric thickness (which is typically close or equal to zero). See “Specifying the constitutive thickness” in “Defining the cohesive element’s initial geometry,” Section 32.5.4, for a discussion on how to modify the constitutive thickness.
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The nominal traction stress vector, , consists of three components (two components in two-dimensional problems): $t _ { n } , t _ { s } ,$ , and (in three-dimensional problems) $t _ { t }$ , which represent the normal (along the local 3-direction in three dimensions and along the local 2-direction in two dimensions) and the two shear tractions (along the local 1- and 2-directions in three dimensions and along the local 1-direction in two dimensions), respectively. The corresponding separations are denoted by $\delta _ { n } , \delta _ { s }$ , and $\delta _ { t }$ . Denoting by $T _ { o }$ the original thickness of the cohesive element, the nominal strains can be defined as
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$$
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\varepsilon_ {n} = \frac {\delta_ {n}}{T _ {o}}, \varepsilon_ {s} = \frac {\delta_ {s}}{T _ {o}}, \varepsilon_ {t} = \frac {\delta_ {t}}{T _ {o}}.
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$$
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The elastic behavior can then be written as
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$$
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\mathbf {t} = \left\{ \begin{array}{l} t _ {n} \\ t _ {s} \\ t _ {t} \end{array} \right\} = \left[ \begin{array}{l l l} E _ {n n} & E _ {n s} & E _ {n t} \\ E _ {n s} & E _ {s s} & E _ {s t} \\ E _ {n t} & E _ {s t} & E _ {t t} \end{array} \right] \left\{ \begin{array}{l} \varepsilon_ {n} \\ \varepsilon_ {s} \\ \varepsilon_ {t} \end{array} \right\} = \mathbf {E} \boldsymbol {\varepsilon}.
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$$
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The elasticity matrix provides fully coupled behavior between all components of the traction vector and separation vector and can depend on temperature and/or field variables. Set the off-diagonal terms in the elasticity matrix to zero if uncoupled behavior between the normal and shear components is desired.
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<!-- source-page: 863 -->
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Optionally, for the uncoupled traction behavior a compression factor can be specified; this ensures that the compressive stiffness is equal to the specified factor times the tensile stiffness, $E _ { n n }$ . This factor affects only the traction response for separation in the normal direction; the shear behavior is not affected.
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Input File Usage: Use the following option to define uncoupled traction-separation behavior:
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\*ELASTIC, TYPE=TRACTION
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Use the following option to define uncoupled traction-separation behavior with a compression factor:
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\*ELASTIC, TYPE=TRACTION, COMPRESSION FACTOR=f
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Use the following option to define coupled traction-separation behavior:
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\*ELASTIC, TYPE=COUPLED TRACTION
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Abaqus/CAE Usage: Use the following option to define uncoupled traction-separation behavior:
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Property module: material editor: Mechanical→Elasticity→Elastic:
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# Type: Traction
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Use the following option to define coupled traction-separation behavior:
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Property module: material editor: Mechanical→Elasticity→Elastic:
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# Type: Coupled Traction
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Specifying a compression factor for uncoupled traction-separation behavior is not supported in Abaqus/CAE.
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# Interpretation of material properties
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The material parameters, such as the interfacial elastic stiffness, for a traction-separation model can be better understood by studying the equation that represents the displacement of a truss of length $L ,$ elastic stiffness $E ,$ and original area A, due to an axial load P:
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$$
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\delta = \frac {P L}{A E}.
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$$
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This equation can be rewritten as
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$$
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\delta = \frac {S}{K},
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$$
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where $S = P / A$ is the nominal stress and $K = E / L$ is the stiffness that relates the nominal stress to the displacement. Likewise, the total mass of the truss, assuming a density $\rho ,$ is given by
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$$
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M = \rho A L = \bar {\rho} A.
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$$
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The above equations suggest that the actual length L may be replaced with 1.0 (to ensure that the strain is the same as the displacement) if the stiffness and the density are appropriately reinterpreted. In particular, the stiffness is $K = ( E / L )$ and the density is $\bar { \rho } = \left( \rho L \right)$ , where the true length of the truss is used in these equations. The density represents mass per unit area instead of mass per unit volume.
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<!-- source-page: 864 -->
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These ideas can be carried over to a cohesive layer of initial thickness $T _ { c }$ . If the adhesive material has stiffness $E _ { a }$ and density $\rho _ { c } ,$ , the stiffness of the interface (relating the nominal traction to the nominal strain) is given by ${ \cal E } _ { c } = ( { \cal E } _ { a } / T _ { c } ) T _ { o }$ and the density of the interface is given by $\bar { \rho _ { c } } ~ = ~ \left( \rho _ { c } T _ { c } \right)$ . As discussed earlier, the default choice of the constitutive thickness $T _ { o }$ for modeling the response in terms of traction versus separation is 1.0 regardless of the actual thickness of the cohesive layer. With this choice, the nominal strains are equal to the corresponding separations. When the constitutive thickness of the cohesive layer is “artificially” set to 1.0, ideally you should specify $E _ { c }$ and $\bar { \rho _ { c } }$ (if needed) as the material stiffness and density, respectively, as calculated with the true thickness of the cohesive layer.
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The above formulae provide a recipe for estimating the parameters required for modeling the traction-separation behavior of an interface in terms of the material properties of the bulk adhesive material. As the thickness of the interface layer tends to zero, the above equations imply that the stiffness, $E _ { c } { _ { \mathrm { : } } }$ , tends to infinity and the density, $\bar { \rho _ { c } }$ , tends to zero. This stiffness is often chosen as a penalty parameter. A very large penalty stiffness is detrimental to the stable time increment in Abaqus/Explicit and may result in ill-conditioning of the element operator in Abaqus/Standard. Recommendations for the choice of the stiffness and density of an interface for an Abaqus/Explicit analysis such that the stable time increment is not adversely affected are provided in “Stable time increment in Abaqus/Explicit” in “Modeling with cohesive elements,” Section 32.5.3.
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# Modeling rate-dependent traction-separation behavior in Abaqus/Explicit
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Time domain viscoelasticity can be used in Abaqus/Explicit to model rate-dependent behavior of cohesive elements with traction-separation elasticity. The evolution equation for the normal and two shear nominal tractions take the form:
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$$
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t _ {n} (t) = t _ {n} ^ {0} (t) + \int_ {0} ^ {t} \dot {k} _ {R} (s) t _ {n} ^ {0} (t - s) d s,
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$$
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$$
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t _ {s} (t) = t _ {s} ^ {0} (t) + \int_ {0} ^ {t} \dot {g} _ {R} (s) t _ {s} ^ {0} (t - s) d s,
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$$
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$$
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t _ {t} (t) = t _ {t} ^ {0} (t) + \int_ {0} ^ {t} \dot {g} _ {R} (s) t _ {t} ^ {0} (t - s) d s,
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$$
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where $t _ { n } ^ { 0 } ( t ) , t _ { s } ^ { 0 } ( t )$ , and $t _ { t } ^ { 0 } ( t )$ are the instantaneous nominal tractions at time t in the normal and the two local shear directions, respectively. The functions $g _ { R } ( t )$ and $k _ { R } ( t )$ represent the dimensionless shear and normal relaxation moduli, respectively. See “Defining viscoelastic behavior for traction-separation elasticity in Abaqus/Explicit” in “Time domain viscoelasticity,” Section 22.7.1, for additional details and usage information.
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You can also combine time domain viscoelasticity with the models for progressive damage and failure described in the next sections. This combination allows modeling rate-dependent behavior both during the initial elastic response (prior to damage initiation), as well as during damage progression.
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<!-- source-page: 865 -->
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# Damage modeling
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Both Abaqus/Standard and Abaqus/Explicit allow modeling of progressive damage and failure in cohesive layers whose response is defined in terms of traction-separation. By comparison, only Abaqus/Explicit allows modeling of progressive damage and failure for cohesive elements modeled with conventional materials (“Defining the constitutive response of cohesive elements using a continuum approach,” Section 32.5.5). Damage of the traction-separation response is defined within the same general framework used for conventional materials (see “Progressive damage and failure,” Section 24.1.1). This general framework allows the combination of several damage mechanisms acting simultaneously on the same material. Each failure mechanism consists of three ingredients: a damage initiation criterion, a damage evolution law, and a choice of element removal (or deletion) upon reaching a completely damaged state. While this general framework is the same for traction-separation response and conventional materials, many details of how the various ingredients are defined are different. Therefore, the details of damage modeling for traction-separation response are presented below.
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The initial response of the cohesive element is assumed to be linear as discussed above. However, once a damage initiation criterion is met, material damage can occur according to a user-defined damage evolution law. Figure 32.5.6–1 shows a typical traction-separation response with a failure mechanism. If the damage initiation criterion is specified without a corresponding damage evolution model, Abaqus will evaluate the damage initiation criterion for output purposes only; there is no effect on the response of the cohesive element (i.e., no damage will occur). The cohesive layer does not undergo damage under pure compression.
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<details>
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<summary>line</summary>
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| separation | traction |
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| ---------- | -------- |
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| δ_n^o(δ_s^o, δ_t^o) | t_n^o(t_s^o, t_t^o) |
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| δ_n^f(δ_s^f, δ_t^f) | 0 |
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</details>
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Figure 32.5.6–1 Typical traction-separation response.
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<!-- source-page: 866 -->
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# Damage initiation
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As the name implies, damage initiation refers to the beginning of degradation of the response of a material point. The process of degradation begins when the stresses and/or strains satisfy certain damage initiation criteria that you specify. Several damage initiation criteria are available and are discussed below. Each damage initiation criterion also has an output variable associated with it to indicate whether the criterion is met. A value of 1 or higher indicates that the initiation criterion has been met (see “Output,” for further details). Damage initiation criteria that do not have an associated evolution law affect only output. Thus, you can use these criteria to evaluate the propensity of the material to undergo damage without actually modeling the damage process (i.e., without actually specifying damage evolution).
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In the discussion below, $t _ { n } ^ { o } , t _ { s } ^ { o }$ , and $t _ { t } ^ { o }$ represent the peak values of the nominal stress when the deformation is either purely normal to the interface or purely in the first or the second shear direction, respectively. Likewise, $\varepsilon _ { n } ^ { o } , \varepsilon _ { s } ^ { o }$ , and $\varepsilon _ { t } ^ { o }$ represent the peak values of the nominal strain when the deformation is either purely normal to the interface or purely in the first or the second shear direction, respectively. With the initial constitutive thickness $T _ { o } = 1$ , the nominal strain components are equal to the respective components of the relative displacement— $- \delta _ { n } , \delta _ { s }$ , and $\delta _ { t }$ —between the top and bottom of the cohesive layer. The symbol used in the discussion below represents the Macaulay bracket with the usual interpretation. The Macaulay brackets are used to signify that a pure compressive deformation or stress state does not initiate damage.
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# Maximum nominal stress criterion
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Damage is assumed to initiate when the maximum nominal stress ratio (as defined in the expression below) reaches a value of one. This criterion can be represented as
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$$
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\max \biggl \{\frac {\langle t _ {n} \rangle}{t _ {n} ^ {o}}, \frac {t _ {s}}{t _ {s} ^ {o}}, \frac {t _ {t}}{t _ {t} ^ {o}} \biggr \} = 1.
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$$
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Input File Usage: \*DAMAGE INITIATION, CRITERION=MAXS
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Abaqus/CAE Usage: Property module: material editor: Mechanical→Damage for Traction-Separation Laws→Maxs Damage
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# Maximum nominal strain criterion
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Damage is assumed to initiate when the maximum nominal strain ratio (as defined in the expression below) reaches a value of one. This criterion can be represented as
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$$
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\max \biggl \{\frac {\left\langle \varepsilon_ {n} \right\rangle}{\varepsilon_ {n} ^ {o}}, \frac {\varepsilon_ {s}}{\varepsilon_ {s} ^ {o}}, \frac {\varepsilon_ {t}}{\varepsilon_ {t} ^ {o}} \biggr \} = 1.
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$$
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Input File Usage: \*DAMAGE INITIATION, CRITERION=MAXE
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Abaqus/CAE Usage: Property module: material editor: Mechanical→Damage for Traction-Separation Laws→Maxe Damage
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<!-- source-page: 867 -->
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# Quadratic nominal stress criterion
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Damage is assumed to initiate when a quadratic interaction function involving the nominal stress ratios (as defined in the expression below) reaches a value of one. This criterion can be represented as
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$$
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\left\{\frac {\langle t _ {n} \rangle}{t _ {n} ^ {o}} \right\} ^ {2} + \left\{\frac {t _ {s}}{t _ {s} ^ {o}} \right\} ^ {2} + \left\{\frac {t _ {t}}{t _ {t} ^ {o}} \right\} ^ {2} = 1.
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$$
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Input File Usage: \*DAMAGE INITIATION, CRITERION=QUADS
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Abaqus/CAE Usage: Property module: material editor: Mechanical→Damage for Traction-Separation Laws→Quads Damage
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# Quadratic nominal strain criterion
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Damage is assumed to initiate when a quadratic interaction function involving the nominal strain ratios (as defined in the expression below) reaches a value of one. This criterion can be represented as
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$$
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\left\{\frac {\left\langle \varepsilon_ {n} \right\rangle}{\varepsilon_ {n} ^ {o}} \right\} ^ {2} + \left\{\frac {\varepsilon_ {s}}{\varepsilon_ {s} ^ {o}} \right\} ^ {2} + \left\{\frac {\varepsilon_ {t}}{\varepsilon_ {t} ^ {o}} \right\} ^ {2} = 1.
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$$
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Input File Usage: \*DAMAGE INITIATION, CRITERION=QUADE
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Abaqus/CAE Usage: Property module: material editor: Mechanical→Damage for Traction-Separation Laws→Quade Damage
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# Damage evolution
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The damage evolution law describes the rate at which the material stiffness is degraded once the corresponding initiation criterion is reached. The general framework for describing the evolution of damage in bulk materials (as opposed to interfaces modeled using cohesive elements) is described in “Damage evolution and element removal for ductile metals,” Section 24.2.3. Conceptually, similar ideas apply for describing damage evolution in cohesive elements with a constitutive response that is described in terms of traction versus separation; however, many details are different.
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A scalar damage variable, D, represents the overall damage in the material and captures the combined effects of all the active mechanisms. It initially has a value of 0. If damage evolution is modeled, D monotonically evolves from 0 to 1 upon further loading after the initiation of damage. The stress components of the traction-separation model are affected by the damage according to
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$$
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t _ {n} = \left\{ \begin{array}{l l} (1 - D) \bar {t} _ {n}, & \bar {t} _ {n} \geq 0 \\ \bar {t} _ {n}, & \text { otherwise (no damage to compressive stiffness) }; \end{array} \right.
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$$
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$$
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t _ {s} = (1 - D) \bar {t} _ {s},
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$$
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$$
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t _ {t} = (1 - D) \bar {t} _ {t},
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$$
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<!-- source-page: 868 -->
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where ${ \bar { t } _ { n } } , \bar { t } _ { s }$ and $\overline { { t } } _ { t }$ are the stress components predicted by the elastic traction-separation behavior for the current strains without damage.
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To describe the evolution of damage under a combination of normal and shear deformation across the interface, it is useful to introduce an effective displacement (Camanho and Davila, 2002) defined as
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$$
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\delta_ {m} = \sqrt {\langle \delta_ {n} \rangle^ {2} + \delta_ {s} ^ {2} + \delta_ {t} ^ {2}}.
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$$
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# Mixed-mode definition
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The mode mix of the deformation fields in the cohesive zone quantify the relative proportions of normal and shear deformation. Abaqus uses three measures of mode mix, two that are based on energies and one that is based on tractions. You can choose one of these measures when you specify the mode dependence of the damage evolution process. Denoting by $G _ { n } , G _ { s }$ , and $G _ { t }$ the work done by the tractions and their conjugate relative displacements in the normal, first, and second shear directions, respectively, and defining $G _ { T } = G _ { n } + G _ { s } + G _ { t }$ , the mode-mix definitions based on energies are as follows:
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$$
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m _ {1} = \frac {G _ {n}}{G _ {T}},
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$$
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$$
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m _ {2} = \frac {G _ {s}}{G _ {T}},
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$$
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$$
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m _ {3} = \frac {G _ {t}}{G _ {T}}.
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$$
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Clearly, only two of the three quantities defined above are independent. It is also useful to define the quantity $G _ { S } ~ = ~ G _ { s } + G _ { t }$ to denote the portion of the total work done by the shear traction and the corresponding relative displacement components. As discussed later, Abaqus requires that you specify material properties related to damage evolution as functions of $m _ { 2 } + m _ { 3 } ( = G _ { S } / G _ { T } )$ (or, equivalently, $1 - m _ { 1 } )$ and $m _ { 3 } / ( m _ { 2 } + m _ { 3 } ) ( = G _ { t } / G _ { S } )$ .
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Abaqus computes the energy quantities described above either based on the current state of deformation (nonaccumulative measure of energy) or based on the deformation history (accumulative measure of energy) at an integration point. The former approach is useful in mixed-mode simulations where the primary energy dissipation mechanism is associated with the creation of new surfaces due to failure in the cohesive zone. Such problems are typically adequately described utilizing the methods of linear elastic fracture mechanics. The latter approach provides an alternate way of defining the mode-mix and may be useful in situations where other significant dissipation mechanisms also govern the overall structural response.
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The corresponding definitions of the mode mix based on traction components are given by
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$$
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\phi_ {1} = \left(\frac {2}{\pi}\right) \tan^ {- 1} \left(\frac {\tau}{\langle t _ {n} \rangle}\right),
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$$
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<!-- source-page: 869 -->
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$$
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\phi_ {2} = \left(\frac {2}{\pi}\right) \tan^ {- 1} \left(\left| \frac {t _ {t}}{t _ {s}} \right|\right),
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$$
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where $\tau = \sqrt { t _ { s } ^ { 2 } + t _ { t } ^ { 2 } }$ is a measure of the effective shear traction. The angular measures used in the above definition (before they are normalized by the factor $2 / \pi )$ are illustrated in Figure 32.5.6–2.
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<details>
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<summary>text_image</summary>
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normal
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tn
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t
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φ1
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φ2
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τ
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ts
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Shear 1
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tt
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Shear 2
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</details>
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Figure 32.5.6–2 Mode mix measures based on traction.
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Input File Usage: Use the following option to use the mode-mix definition based on nonaccumulated energies:
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\*DAMAGE EVOLUTION, MODE MIX RATIO=ENERGY
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<!-- source-page: 870 -->
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Use the following option to use the mode-mix definition based on accumulated energies:
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\*DAMAGE EVOLUTION, MODE MIX RATIO=ACCUMULATED ENERGY
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Use the following option to use the mode-mix definition based on tractions:
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\*DAMAGE EVOLUTION, MODE MIX RATIO=TRACTION
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Abaqus/CAE Usage: Property module: material editor: Mechanical→Damage for Traction-Separation Laws→Quade Damage, Maxe Damage, Quads Damage, or Maxs Damage: Suboptions→Damage Evolution: Mode mix ratio: Energy or Traction
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Specifying a mode-mix definition based on accumulated energies is not supported in Abaqus/CAE.
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# Comparison of mixed-mode definitions
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The mode-mix ratios defined in terms of the different energy quantities and tractions can be quite different in general. The following examples illustrate this point. In terms of energies a deformation in the purely normal direction is one for which $G _ { n } \neq 0$ and $G _ { s } = G _ { t } = 0$ , irrespective of the values of the normal and the shear tractions. In particular, for a material with coupled traction-separation behavior both the normal and shear tractions may be nonzero for a deformation in the purely normal direction. For this case the definition of mode mix based on energies would indicate a purely normal deformation, while the definition based on tractions would suggest a mix of both normal and shear deformation.
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When the mode mix is defined based on accumulated energies, an artificial path-dependence may be introduced in the mixed-mode behavior that may not be consistent, for example, with predictions that are based on linear elastic fracture mechanics. Therefore, if an interface is first loaded purely in the normal deformation mode, unloaded, and subsequently loaded in a purely shear deformation mode, the mode-mix ratios based on accumulated energies at the end of the above deformation path evaluate to (assuming the shear deformation to be in the local-1 direction only) $G _ { n } \neq 0$ and $G _ { s } \neq 0$ . On the other hand, the mode-mix ratios based on nonaccumulated energies evaluate to $G _ { n } = 0$ and $G _ { s } \neq 0$ at the end of the above deformation path.
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# Damage evolution definition
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There are two components to the definition of the evolution of damage. The first component involves specifying either the effective displacement at complete failure, $\delta _ { m } ^ { f }$ , relative to the effective displacement at the initiation of damage, $\delta _ { m } ^ { o }$ ; or the energy dissipated due to failure, $G ^ { C }$ (see Figure 32.5.6–3). The second component to the definition of damage evolution is the specification of the nature of the evolution of the damage variable, D, between initiation of damage and final failure. This can be done by either defining linear or exponential softening laws or specifying D directly as a tabular function of the effective displacement relative to the effective displacement at damage initiation. The material data described above will in general be functions of the mode mix, temperature, and/or field variables.
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Figure 32.5.6–4 is a schematic representation of the dependence of damage initiation and evolution on the mode mix, for a traction-separation response with isotropic shear behavior. The figure shows the
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