A shifted energy barrier approach for phase-field modeling of tensile-dominated brittle fracture

Let $\Omega\subset\mathbb{R}^{n}$ ($n\in\{1,2,3\}$) be an open and bounded domain representing a solid body, with its external boundary denoted by $\partial\Omega$. An internal sharp crack is represented by a lower-dimensional set $\mathcal{C}$ embedded in the body. Following 22, Griffith’s brittle fracture theory can be formulated as an energy minimization problem.

Neglecting body forces and surface tractions for simplicity, and considering a time-dependent Dirichlet boundary condition $\mathbf{u}(\mathbf{x},t)=\bar{\mathbf{u}}(\mathbf{x},t)$ on $\partial_{D}\Omega$, the total potential energy of the cracked body is written as

where $G_{c}>0$ is the critical energy release rate, and $\mathbf{u}$ is the kinematically admissible displacement field. The infinitesimal strain tensor is defined as $\bm{\varepsilon}(\mathbf{u})=\frac{1}{2}(\nabla\mathbf{u}+(\nabla\mathbf{u})^{T})$.

For an isotropic linear elastic material, the elastic strain energy density is

where $\kappa$ and $\mu$ are the bulk and shear moduli, respectively. The strain tensor is decomposed into spherical and deviatoric parts as

where $\mathbf{I}$ is the second-order identity tensor. The Cauchy stress tensor follows from the elastic potential:

Within this variational setting, the quasi-static evolution of $\mathbf{u}$ and $\mathcal{C}$ at time $t$ is obtained by minimizing Equation 1 subject to the crack irreversibility constraint:

Tracking the discontinuous crack surface $\mathcal{C}$ in a complex domain is computationally demanding. The phase-field method avoids explicit crack tracking by introducing a continuous scalar damage variable $d(\mathbf{x},t)\in[0,1]$, where $d=0$ denotes the intact state and $d=1$ denotes the fully broken state. The sharp crack is regularized over a finite width controlled by the regularization length $l_{0}>0$.

Among phase-field formulations, the $\mathrm{AT}_{1}$ model is useful for studying crack nucleation because it provides an elastic limit before damage initiation. The Griffith surface energy in Equation 1 is approximated by the $\mathrm{AT}_{1}$ crack density functional (49; 58):

To reduce non-physical crack growth under compression, a tension–compression split of the elastic energy is commonly introduced. Following 3, the strain energy density is decomposed into an active part $\psi_{e}^{+}$ and a passive part $\psi_{e}^{-}$:

with

where $\langle x\rangle_{\pm}=(x\pm|x|)/2$ are the Macaulay brackets. The corresponding stress contributions are denoted by $\bm{\sigma}_{0}^{+}=\partial\psi_{e}^{+}/\partial\bm{\varepsilon}$ and $\bm{\sigma}_{0}^{-}=\partial\psi_{e}^{-}/\partial\bm{\varepsilon}$. 60 established that eliminating spurious residual shear stresses at full damage is essential to prevent unphysical traction transmission across separated crack faces, a physical requirement strictly satisfied by Amor’s decomposition.

Using the degradation function $g(d)=(1-d)^{2}$, the regularized $\mathrm{AT}_{1}$ energy functional is

For a quasi-static process at time $t\in[0,T]$, the coupled state $(\mathbf{u}(\mathbf{x},t),d(\mathbf{x},t))$ is obtained from the constrained local minimization problem

where

and

Here, $\mathcal{D}_{t}$ enforces the irreversibility of the phase field by requiring the current damage field to be no smaller than its previous maximum value.

To clarify why the classical $\mathrm{AT}_{1}$ model cannot independently prescribe material strength, we examine the onset of crack nucleation in a homogeneous intact body. The strong form of the phase-field evolution follows from the first-order optimality conditions of Equation 10. Taking the first variation of $\mathcal{E}_{\mathrm{AT}_{1}}^{*}$ with respect to $d$ gives the Karush–Kuhn–Tucker (KKT) conditions

together with the complementarity condition

When damage grows, the constraint becomes active and the bracketed term vanishes:

For an initially intact homogeneous body under monotonically increasing loading, we have $d=0$, $\nabla d=\mathbf{0}$, and thus $\nabla^{2}d=0$ before nucleation. At the onset of damage, the active condition in Equation 15 gives

With $g(d)=(1-d)^{2}$, one has $g^{\prime}(d)=-2(1-d)$ and $g^{\prime}(0)=-2$. Therefore, the critical active elastic energy density required for damage initiation is

This coupling between strength, toughness, and the regularization length limits the direct use of the classical $\mathrm{AT}_{1}$ model for materials whose failure is governed by a prescribed tensile strength. The following section introduces a shifted driving force to align the phase-field nucleation condition with the Rankine criterion while retaining the $\mathrm{AT}_{1}$ fracture energy.

Let $\bm{\sigma}_{0}(\bm{\varepsilon})$ denote the effective undamaged elastic Cauchy stress tensor. The principal stresses of $\bm{\sigma}_{0}$ are ordered as $\sigma_{1}\geq\sigma_{2}\geq\sigma_{3}$. In the present formulation, crack nucleation is linked to the effective stress state and is assumed to occur when the maximum effective principal tensile stress reaches the uniaxial tensile strength $\sigma_{t}$:

To incorporate the stress-based Rankine criterion into the energy-driven $\mathrm{AT}_{1}$ framework, the tensile strength $\sigma_{t}$ is mapped to an equivalent active elastic energy threshold, denoted by $\psi_{cs}$.

In general, however, a strength surface cannot be expressed as a fixed energy or strain threshold; equivalent energy or strain thresholds exist only in special cases such as uniaxial tension (32). The threshold $\psi_{cs}$ is therefore not a fixed energy level but is made state-dependent: since the active elastic energy of a general multiaxial state depends on the full strain tensor, $\psi_{cs}$ is obtained by scaling the current effective stress state to the Rankine surface.

For a given undamaged elastic state with $\sigma_{1}>0$, consider a virtual proportional scaling of the effective stress tensor until the condition $\sigma_{1}=\sigma_{t}$ is reached. The corresponding scaling factor is

Since the elastic strain energy is quadratic in stress under linear elasticity, the active elastic energy scales with $\eta^{2}$. The Rankine-equivalent critical active energy density is defined as

The infinite threshold assigned for $\sigma_{1}\leq 0$ precludes tensile crack nucleation under compressive effective stress states; its finite numerical surrogate is discussed in Remark 6.

As shown in Section 2.3, the classical $\mathrm{AT}_{1}$ model contains the intrinsic energy barrier $\psi_{c}=3G_{c}/(16l_{0})$. Damage starts when the active elastic energy $\psi_{e}^{+}$ reaches this barrier. Since $\psi_{c}$ is controlled by $G_{c}$ and $l_{0}$, it does not generally coincide with the energy level associated with a prescribed tensile strength. To link crack nucleation to the Rankine criterion, the damage driving term is shifted so that the onset condition becomes $\psi_{e}^{+}=\psi_{cs}$.

We define the shifted active energy density as

This shift modifies the damage driving force, while the $\mathrm{AT}_{1}$ crack density is kept unchanged.

Replacing $\psi_{e}^{+}$ by $\widehat{\psi}_{e}^{+}$ in the active damage equation gives

At crack nucleation in an initially intact homogeneous state, $d=0$, $\nabla d=\mathbf{0}$, and $\nabla^{2}d=0$. Using $g^{\prime}(0)=-2$, Equation 22 gives

Substituting Equation 21 and $\psi_{c}=3G_{c}/(16l_{0})$ yields

and therefore

Thus, the shifted driving force changes the $\mathrm{AT}_{1}$ nucleation condition from the intrinsic barrier $\psi_{c}$ to the Rankine-equivalent threshold $\psi_{cs}$. For $\sigma_{1}>0$, substituting Equation 20 into Equation 25 gives

which, for $\psi_{e}^{+}>0$ with $\sigma_{t}>0$ and $\sigma_{1}>0$, reduces to $\sigma_{1}=\sigma_{t}$. The onset condition Equation 25 is therefore equivalent to the Rankine criterion Equation 18 evaluated at the current effective stress state.

After damage initiation, the current threshold $\psi_{cs}$ is replaced by a frozen history threshold, which will be introduced in Section 4. This avoids changes of the activated damage resistance during unloading or non-proportional changes in the stress state.

The classical variational phase-field formulation of Section 2 derives from a single energy functional, so that the stress response and the damage driving force are governed by one common potential; the formulation is in this sense variationally consistent, with a self-adjoint structure (22; 8; 49; 61). In the present shifted formulation this is no longer the case: because the Rankine-equivalent threshold $\psi_{cs}$ depends on the strain through the effective stress, the equilibrium and damage equations cannot be obtained from a common potential. We therefore derive them from a restricted variational principle (53; 54), in which a state-dependent quantity is held fixed during the variation and its dependence on the primary fields is restored only afterwards. As analyzed by 20, the operator associated with such formulations is in general not self-adjoint, so they do not constitute genuine minimization principles. In this work, the restricted variational principle is used only as a systematic derivation device, the physical admissibility of the resulting evolution being established separately in Section 4.

In the present formulation, the state-dependent quantity treated in this restricted sense is the Rankine-equivalent threshold $\psi_{cs}=\psi_{cs}(\bm{\varepsilon}(\mathbf{u}))$. At a given load step, $\psi_{cs}$ is evaluated from the effective stress state used to define the damage-initiation threshold and is then kept fixed during the variations with respect to the primary fields. We introduce the restricted shifted functional

The restricted shifted functional is used only to derive the mechanical equilibrium equation and the shifted phase-field conditions under this prescribed-threshold restriction; it is not the Helmholtz free energy entering the thermodynamic dissipation inequality. The passive elastic energy $\psi_{e}^{-}$ is retained in the displacement variation to preserve the standard compressive elastic response, but does not contribute to the phase-field driving force because it is not degraded by $g(d)$.

Consider first the variation with respect to the displacement field. Since $\psi_{cs}$ depends on the strain through the effective stress, a full variation of $\mathcal{E}_{\mathrm{shift}}$ would yield the stress response

in which the additional term $-g(d)\,\partial\psi_{cs}/\partial\bm{\varepsilon}$ would modify the standard degraded elastic response and is not part of the constitutive assumptions of the present model. In the restricted variation, $\psi_{cs}$ (together with the constant $\psi_{c}$) is held fixed, so its derivative with respect to strain vanishes and the stress reduces to

Thus the restricted variation lets the shifted threshold modify the phase-field driving force while leaving the degraded elastic stress unchanged. This is precisely the mechanism by which the restricted variational principle provides a route to a hybrid formulation (2): the prescribed threshold drives the damage evolution without altering the mechanical equilibrium. The use of a non-energetic, strength-based contribution to the damage driving force is shared by recent phase-field formulations that introduce material strength as an independent input (35; 37; 40); here it is realized through the restricted variation of the shifted threshold while the standard $\mathrm{AT}_{1}$ stress is retained.

Taking the restricted variation with respect to the phase-field variable $d$ gives

The irreversible damage evolution is then governed by the KKT (loading–unloading) conditions associated with the irreversibility constraint $\dot{d}\geq 0$ and the driving force in Equation 29:

These conditions are stated as a complementarity (variational-inequality) problem rather than as the optimality conditions of a global minimization. They coincide with the optimality conditions of the bound-constrained phase-field subproblem solved at held-fixed threshold (Section 6), and the driving force entering them is the one obtained thermodynamically from the dissipative microforce balance in Section 4, where the associated damage dissipation is shown to be non-negative.

For an initially intact homogeneous state, $d=0$ and $\nabla^{2}d=0$, so the active complementarity condition Equation 30c reduces to Equation 24, and the nucleation condition $\psi_{e}^{+}=\psi_{cs}$ of Equation 25 is recovered. The restricted variational principle therefore reproduces the shifted nucleation condition while preserving the standard degraded elastic stress response in Equation 28.

The recoverable Helmholtz free-energy density is taken as the standard $\mathrm{AT}_{1}$ density, i.e. the integrand of Equation 9,

so that $\mathcal{E}_{\mathrm{AT}_{1}}^{*}=\int_{\Omega}\Psi_{0}\,d\Omega$. Since the Rankine-equivalent threshold $\psi_{cs}$ is not included in $\Psi_{0}$, the recoverable stress remains the degraded elastic stress already given in Equation 28.

Because $\Psi_{0}$ depends on $d$ and $\nabla d$, the phase-field variable is treated as a gradient-type internal variable. A scalar microforce $\pi$, conjugate to $\dot{d}$, and a vector microstress $\bm{\xi}$, conjugate to $\nabla\dot{d}$, are introduced. The local isothermal free-energy imbalance is written as

Application of the Coleman–Noll procedure (13) to the recoverable rates gives

The remaining contribution to the dissipation is associated with the scalar microforce conjugate to $\dot{d}$. We define

The coefficient of $\dot{d}$ in Equation 34 is not set to zero. Crack irreversibility restricts the admissible damage rates to $\dot{d}\geq 0$. Therefore, the remaining scalar microforce contribution is dissipative.

In the absence of external microforces associated with $d$, the local microforce balance is

Combining Equations 33, 34 and 35 gives

For the standard $\mathrm{AT}_{1}$ density, the energetic terms in Equation 36 recover the classical phase-field balance derived in Section 2.3. Consequently, a Rankine-based modification of the phase-field evolution must enter through the dissipative scalar microforce, rather than through the recoverable free energy.

Before damage initiation, the threshold controlling the shifted response is the current Rankine-equivalent value $\psi_{cs}$, because the intended onset condition is $\psi_{e}^{+}=\psi_{cs}$. After damage initiation, continuous updating of $\psi_{cs}$ from the current stress state would make the activated resistance vary during unloading or non-proportional loading. The threshold entering the dissipative microforce is therefore frozen after initiation:

where $t_{\mathrm{ini}}(\mathbf{x})$ is the local damage-initiation time, defined as the first instant at which $d(\mathbf{x},t)>0$. Accordingly, $\mathcal{H}_{cs}$ follows the current Rankine-equivalent threshold while the material point is intact and is held fixed at its initiation value once damage has started, so that $\dot{\mathcal{H}}_{cs}=0$ for $d>0$. In this sense $\mathcal{H}_{cs}$ is the central quantity of the shifted formulation: before initiation it fixes the strength-controlled nucleation condition $\psi_{e}^{+}=\mathcal{H}_{cs}$, and once frozen it sets the resistance to subsequent damage growth. Because it is frozen rather than load-following, the Rankine shift enters as a well-defined dissipative resistance, whose thermodynamic admissibility is established in Section 4.3.

The dissipative scalar microforce associated with the Rankine-based shift is prescribed as

where the intrinsic $\mathrm{AT}_{1}$ barrier $\psi_{c}$ is given in Equation 17. The factor $-g^{\prime}(d)$ makes the additional resistance enter the phase-field balance with the same degradation-weighted structure as the active elastic driving contribution.

Substitution of Equation 38 into Equation 36 gives the phase-field balance associated with active damage growth:

The balance equation in Equation 39 has the same residual form as the equation obtained from the restricted variation in Equation 29, with $\psi_{cs}$ replaced by the frozen threshold $\mathcal{H}_{cs}$ after initiation. The complete irreversible evolution remains governed by the constrained phase-field problem and the KKT conditions stated in Section 3.3. The role of the thermodynamic argument is to identify the shifted term as a dissipative microforce contribution and to examine the sign of the associated damage dissipation.

Before damage initiation, $\mathcal{H}_{cs}=\psi_{cs}$. For an initially intact state, Equation 39 reduces to the same nucleation condition $\psi_{e}^{+}=\psi_{cs}$. As shown in Sections 3.2 and 3.3, this condition is equivalent to the Rankine criterion on the tensile branch, where the Rankine-equivalent threshold is finite. The dissipative-microforce formulation therefore preserves the Rankine-controlled nucleation condition derived from the restricted variational formulation.

Using Equation 38, the local damage dissipation becomes

Because irreversibility requires $\dot{d}\geq 0$, the case $\dot{d}=0$ gives zero dissipation. For all admissible damage growth processes, non-negative dissipation is guaranteed if

where $g(d)=(1-d)^{2}$ has been used. Since $\mathcal{H}_{cs}$ is the frozen value of $\psi_{cs}$ at initiation, Equation 41 requires $\psi_{cs}(t_{\mathrm{ini}})\geq\psi_{c}$. For uniaxial tension, the Rankine-equivalent threshold reduces to the constant value $\psi_{cs}=\sigma_{t}^{2}/(2E)$, as shown in the one-dimensional analysis of Section 5 (see Equation 44). Together with $\psi_{c}=3G_{c}/(16l_{0})$, the admissibility condition gives

The bound in Equation 42 is not a mesh-size requirement, but the condition under which the Rankine-based shift can be interpreted as a non-negative dissipative resistance. If the Rankine-equivalent threshold is lower than the intrinsic $\mathrm{AT}_{1}$ barrier, the shift reduces the resistance to damage growth, and the sign of the dissipation in Equation 40 is not guaranteed.

Consider a bar under uniaxial tension with axial strain $\varepsilon\geq 0$ and Young’s modulus $E$. For $n=1$, the deviatoric strain vanishes, $\bm{\varepsilon}_{\mathrm{dev}}=\mathbf{0}$, and Amor’s decomposition in Equation 7 reduces to

where the one-dimensional modulus $E$ plays the role of $\kappa$. Under tension, the effective stress is $\sigma_{0}=E\varepsilon=\sigma_{1}>0$, and the Rankine-equivalent threshold in Equation 20 becomes

which is constant and independent of the strain state. The freezing rule in Equation 37 therefore has no effect on the damage evolution. The threshold equals $\sigma_{t}^{2}/(2E)$ at every intact point under tension and is frozen at the same value wherever damage has initiated, so that

The shift with respect to the intrinsic $\mathrm{AT}_{1}$ barrier $\psi_{c}=3G_{c}/(16l_{0})$ in Equation 17 is measured by the dimensionless parameter

where $\sigma_{c}=\sqrt{3G_{c}E/(8l_{0})}$ is the nucleation stress of the classical $\mathrm{AT}_{1}$ model (Remark 3). The admissibility condition $\mathcal{H}_{cs}\geq\psi_{c}$ in Equations 41 and 42 is equivalent to $\gamma\geq 0$. Thus, $\gamma$ quantifies the amount by which the prescribed strength exceeds the intrinsic $\mathrm{AT}_{1}$ nucleation stress at the given regularization length, and the limit $\gamma\to 0^{+}$ corresponds to the admissibility boundary.

Using $3G_{c}/(8l_{0})=2\psi_{c}$, $3G_{c}l_{0}/4=4\psi_{c}l_{0}^{2}$, and $\mathcal{H}_{cs}-\psi_{c}=\gamma\,\psi_{c}$, the shifted phase-field balance Equation 39 for active damage growth ($\dot{d}>0$) takes the one-dimensional form

Consider a homogeneous state $(\varepsilon,d)$ with $d^{\prime\prime}=0$ under monotonically increasing strain. The intact state $d=0$ is admissible as long as the one-dimensional form of the KKT inequality Equation 30b holds,

Damage therefore initiates at the Rankine limit $\varepsilon_{t}=\sigma_{t}/E$, independently of $l_{0}$, consistent with Section 3.2.

For $\varepsilon\geq\varepsilon_{t}$, the consistency condition Equation 47 gives the damaging branch

The denominator is positive on this branch, since $\tfrac{E}{2}\varepsilon^{2}\geq\mathcal{H}_{cs}=(1+\gamma)\psi_{c}>\gamma\,\psi_{c}$. At $\varepsilon=\varepsilon_{t}$, the denominator equals $\mathcal{H}_{cs}-\gamma\psi_{c}=\psi_{c}$, so that $d=0$ and the elastic and damaging branches connect continuously. Moreover, $d$ in Equation 49 increases monotonically with $\varepsilon$ and tends to $1$ as $\varepsilon\to\infty$, so the irreversibility constraint $\dot{d}\geq 0$ is satisfied along monotone loading.

Introducing the dimensionless effective stress $\tilde{\sigma}=\sigma_{0}/\sigma_{t}=\varepsilon/\varepsilon_{t}\geq 1$ and using $\tfrac{E}{2}\varepsilon^{2}=(1+\gamma)\tilde{\sigma}^{2}\psi_{c}$, the damaging branch and the nominal (degraded) stress $\sigma=(1-d)^{2}E\varepsilon$ become

For $\gamma=0$, where $\sigma_{t}=\sigma_{c}$, Equation 50 reduces to $1-d=\tilde{\sigma}^{-2}$ and $\sigma/\sigma_{c}=\tilde{\sigma}^{-3}$, which is the homogeneous softening response of the classical $\mathrm{AT}_{1}$ model (49).

The peak nominal stress is attained at initiation. Writing $\psi=\tfrac{E}{2}\varepsilon^{2}$, the nominal stress reads $\sigma=\psi_{c}^{2}E\varepsilon/(\psi-\gamma\psi_{c})^{2}$, and differentiation with respect to $\varepsilon$ gives

The homogeneous response therefore softens immediately after initiation, without a hardening plateau, and the peak nominal stress equals the prescribed strength $\sigma_{t}$. A stability analysis of the homogeneous states in the sense of 50 is beyond the scope of this work. Since the branch Equation 50 softens immediately, however, localization at the peak is expected for sufficiently long bars, as in the classical $\mathrm{AT}_{1}$ model (50; 66).

The localized solution corresponds to the ultimate damage profile of a stress-free crack and is constructed following the classical procedure for gradient-damage models (51; 58). Let the localization band occupy $|x|\leq D$, with the crack center at $x=0$. At complete failure the axial stress vanishes, $\sigma=(1-d)^{2}E\varepsilon=0$, so that $\varepsilon=0$ and hence $\psi_{e}^{+}=0$ at every point of the band where $d<1$. The profile is sought symmetric and monotone, with

where the last condition ensures a smooth ($C^{1}$) matching with the undamaged outer region, and the balance Equation 47 holds with equality on the support of $d$. By Equation 45, the frozen threshold takes the same value $\sigma_{t}^{2}/(2E)$ at every point of the band, irrespective of the local initiation time (Remark 8).

Setting $\psi_{e}^{+}=0$ in Equation 47 and introducing $e=1-d$, so that $d^{\prime\prime}=-e^{\prime\prime}$, one obtains for $\gamma>0$ the linear ordinary differential equation

on $x\in[0,D]$, with $e(0)=0$, $e(D)=1$, and $e^{\prime}(D)=0$. For the classical $\mathrm{AT}_{1}$ model, the corresponding equation is $e^{\prime\prime}=-1/(2l_{0}^{2})$. The shift $\mathcal{H}_{cs}-\psi_{c}=\gamma\psi_{c}$ introduces the additional restoring term $\omega^{2}e$, which changes the profile from parabolic to trigonometric type.

The general solution of Equation 53 is

where the particular solution follows from $1/(2l_{0}^{2}\omega^{2})=1/\gamma$. The condition $e(0)=0$ gives $A=1/\gamma$, and the condition $e^{\prime}(D)=0$ gives $B=A\tan\theta$, with $\theta:=\omega D$. Substituting these constants into the remaining condition $e(D)=1$ yields $A/\cos\theta-1/\gamma=1$, and therefore

The half-width of the localization band and the damage profile then admit the closed forms

with $d(x)=0$ for $|x|\geq D$.

The profile Equation 57 satisfies $d(0)=1$ by Equation 55 and decreases monotonically in $|x|$, since $d^{\prime}(x)\propto-\sin(\theta-\omega|x|)\leq 0$ for $\theta-\omega|x|\in[0,\theta]$. Hence $0\leq d\leq 1$ throughout. Outside the band, $d=0$ and $\varepsilon=0$ give $\sigma_{1}=0$, so that $\psi_{cs}=\infty$ by Equation 20 and the KKT inequality Equation 30b is satisfied. At the boundary of the support, the interior limit of the curvature is $d^{\prime\prime}(D^{-})=(1+\gamma)/(2l_{0}^{2})>0$, with a jump to zero across $x=D$. The classical $\mathrm{AT}_{1}$ profile exhibits the same finite curvature jump at the boundary of its support, with $d^{\prime\prime}=1/(2l_{0}^{2})$, and both profiles are compatible with the $H^{1}$ regularity of the phase field.

The localized profile of the classical $\mathrm{AT}_{1}$ model is parabolic (49; 58),

with half-width $2l_{0}$. The shifted profile Equation 57 is of cosine type, and its half-width Equation 56 depends on the shift parameter $\gamma$. Two properties relate the two profiles.

First, the classical profile is recovered in the limit $\gamma\to 0^{+}$. Expanding Equation 55 for small $\gamma$ gives $\theta=\sqrt{2\gamma}\,\left(1-\tfrac{5}{12}\gamma+O(\gamma^{2})\right)$, and therefore

A Taylor expansion of Equation 57 for small $\gamma$, with $\theta$ and $\omega|x|$ both of order $\sqrt{\gamma}$, gives $d(x)\to(1-|x|/(2l_{0}))^{2}$, so the parabolic profile Equation 58 is recovered pointwise at the admissibility boundary. The slope at the crack center behaves consistently: $|d^{\prime}(0^{\pm})|=\sqrt{(\gamma+2)/2}\,/\,l_{0}\to 1/l_{0}$, which matches the kink of the $\mathrm{AT}_{1}$ profile at $x=0$.

Second, the localization band is strictly narrower than the $\mathrm{AT}_{1}$ band for every $\gamma>0$,

To verify Equation 60, note that $D<2l_{0}$ is equivalent to $\theta<\sqrt{2\gamma}$ with $\gamma=(1-\cos\theta)/\cos\theta$, and hence to