Mesh-Independent Framework for the Bidimensional Analysis of CFRP–Concrete Debonding Shear Tests with Discrete Fracture

Acknowledgments

Notation

A_f

cross-sectional; area of CFRP strip;

A_f,Ref.

cross-sectional; area of reference CFRP strip with a width of 50 mm;

${\mathbf{a}}^{\mathit{e}}$

total displacement vector at the nodes of an element;

${\hat{\mathit{a}}}^{\mathit{e}}$

regular displacement vector at the nodes of an element;

${\mathbf{B}}^{\mathit{e}}$

strain-displacement matrix of an element;

$\mathbf{b}$

body forces;

b_c

width of concrete block;

b_f

CFRP width;

${\mathbf{D}}^{\mathit{e}}$

constitutive matrix of an element;

d(·)

incremental variation of (·);

dσ^e

incremental stress within an element;

dΓ

boundary of the discontinuity;

dΩ

boundary of a cracked body;

f_t

concrete tensile strength;

${\hat{\mathit{f}}}^e$

regular external vector force at the regular nodes;

${\tilde{\mathbf{f}}}^e$

enhanced external vector force at the regular nodes;

${\mathbf{f}}_w^e$

external vector force at the additional nodes;

G_f

concrete fracture energy;

${\mathbf{H}}_{{\mathrm{\Gamma }}_d}^e$

diagonal matrix containing the Heaviside function at each degree of freedom;

${\mathcal{H}}_{{\mathrm{\Gamma }}_d}$

Heaviside function;

${\mathbf{K}}_{\mathbf{a}\mathbf{a}}^e$

stiffness matrix of a standard finite element;

${\mathbf{K}}_{\mathbf{a}\mathbf{w}}^e,{\mathbf{K}}_{\mathbf{w}\mathbf{a}}^e={\mathbf{K}}_{\mathbf{w}\mathbf{w}}^e$

enhanced bulk stiffness matrices;

${\mathbf{K}}_p^e$

penalty matrix;

l_b

bonded length of CFRP strip;

$l_{b,\mathrm{Ref}.}$

bonded length of reference CFRP strip with a width of 50 mm;

$l_d^e$

length of the discontinuity inside an element;

l_u

unbonded bond length of CFRP strip;

${\mathbf{M}}_w^e$

rigid body movement matrix composed by evaluating ${\mathbf{M}}_w^{ek}$M_w^ek at each node of the element;

${\mathbf{M}}_w^{ek}$

rigid body movement matrix for node k of the element;

${\mathbf{N}}_w^e$

shape function matrix for the openings of the crack;

$\mathbf{n}$

unit vector normal to the discontinuity;

s

bond−slip at the loaded end of the FRP strip;

s₀

slip factor dependent on β_w and f_t;

s_max

slip corresponding to complete debonding;

$s_{{\tau }_{\max }}$

slip corresponding to the maximum bond shear stress;

${\mathbf{T}}^{\mathit{e}}$

constitutive matrix for the discontinuity;

$\mathbf{t}$

total traction vetor;

${\mathbf{t}}^{\mathit{e}+}$

traction vector within an element;

t_f

thickness of CFRP strip;

$\mathbf{u}(\mathbf{x})$

continuous displacement field approximation within Ω;

${\mathbf{u}}^{\mathit{e}}$

total displacement field approximation;

$\tilde{\mathbf{u}}(\mathbf{x})$

enhanced displacement field;

$\hat{\mathbf{u}}(\mathbf{x})$

regular displacement field;

$[[\mathbf{u}]]$

jump in u;

${\mathbf{w}}^{\mathit{e}}$

nodal jump vector associated with the discontinuity opening;

(xⁱ, yⁱ)

coordinates of the reference node of the discontinuity;

(x^k, y^k)

coordinates of the regular node;

$\alpha$

factor dependent on G_f, s₀, and τ_max;

α₁

factor with a value of 1.5;

β^e

angle of the discontinuity inside an element;

β_w

width factor accounting for the out−of−plane stresses in 2D versus 3D models;

δ(·)

virtual variation in (·);

${\delta }_{{\mathrm{\Gamma }}_d}$

Dirac delta function along the discontinuity;

ɛ

total strain in the body;

$\hat{\mathit{\epsilon }}$

total strain field corresponding to the continuous part of the displacement;

${\hat{\mathit{\epsilon }}}^{\mathit{e}}$

strain field within an element;

τ

FRP–concrete interfacial bond shear stress;

τ_max

maximum bond shear stress at debonding initiation;

Γ_d

crack discontinuity;

${\mathrm{\Gamma }}_d^e$

crack discontinuity belonging to an element;

${\mathrm{\Gamma }}_t$

crack/traction boundary;

Ω

domain of cracked body;

Ω^e

domain of an element in the mesh;

(· · ·)⁺

quantity pertaining to the first (+) subdomain resulting from the Γ_d split;

(· · ·)⁻

quantity pertaining to the second (−) subdomain resulting from the Γ_d split;

(·)

single contraction;

(:)

double contraction;

(· · ·)^s

symmetric part of (· · ·);

[[· · ·]]

a jump in a quantity between the first (+) and second (−) split subdomains;

∇(· · ·)

gradient of a scalar − valued differentiable function (· · ·); and

⊗

dyadic product.

References