A blended approach for evolving phase fields using peridynamics: Cyclic loading in quasi-brittle fracture

Abstract

A field theory is presented for predicting damage and fracture in quasi brittle materials incorporating effects of irreversible (plastic) deformation as well as elastic moduli that soften with damage. The new observation made here is that material degradation models consistent with plastic dissipation can be described by a two-point history-dependent phase field. This approach blends a two-point phase field with the deformation evolving according to Newton's second law by way of a nonlocal constitutive law. Here the nonlocality is in both space and time. The strain is given by an additive decomposition of elastic strain and irreversible strain. The stress-strain behavior is described by a strength envelope and a family of unloading laws based on damage and plasticity with elastic moduli that degrade in coordination with the accumulation of irreversible strain. The material displacement field is uniquely determined by the initial boundary value problem. The theory satisfies energy balance, with positive energy dissipation rate in accordance with the laws of thermodynamics. The fracture energy of flat cracks is recovered directly from the model and is the product of energy release rate and the crack area, moreover this formula is independent of the length scale of non-locality. The formulation delivers a mesh free method for predicting crack patterns and simulations show quantitative and qualitative agreement with experiments, including hysteresis and damage associated with three-point bending tests on concrete and size effects for quasi-brittle materials.

Figures (19)

• Figure 1: (a) Tensile stress vs strain curve sketched for bond strain with plastic yielding only after peak stress is reached, i.e., $S_t^Y=S_t^C$. Blue lines are admissible stress strain curves in compression and green lines are admissible stress strain curves in tension. These broken lines foliate the strength domain and are admissible constitutive laws for admissible bond stress-strain pairs $(\sigma,S)$. Stress-strain pairs on the strength envelope of unloading laws $g'(r^\ast)/\sqrt{|\mathbf{y}-\mathbf{x}|/L}$ given by dashed curve. The strain is reversible and elastic with constant bond stiffness until $S^*=S_t^Y=S_t^C$. For $S_t^C\leq S^*\leq S_t^F$ then bonds exhibit plasticity and elastic softening. The bond stiffness is proportional to the phase field and corresponding unloading laws are the broken green-blue lines. The plastic tensile strain is $P^*$ given by \ref{['damageplastique']} and is the $S$ axis intercept of the unloading law. The stiffness decreases from $\overline{\mu}=\partial_{r}g(r_t^C)/r_t^C$ smoothly to $0$. The bond offers zero tensile stiffness for bonds broken in tension (red), see Definition \ref{['def: Second']}. However the bond continues to elastically resist negative strains, see blue unloading line connected to red line. (b) Stress versus strain curve sketched with plastic yielding before peak stress now given for compression loading. In this figure the sign of compressive strain and stress are reversed.
• Figure 2: $\gamma(t,\mathbf{y},\mathbf{x},\mathbf{u})$ is one for $S^\ast\leq S_t^Y$ then decays to zero for $S_t^Y<S^\ast< S_t^F$. It is evident from the graph that $\gamma$ is Lipchitz continuous in $S^\ast$.
• Figure 3: Domain $\Omega$ with prescribed Dirichlet data on $\Omega_D^\epsilon$. The union is denoted by $\Omega^*$
• Figure 4: (a) A bond that degrades in tension and compression undergoing stress-strain cycling plotted in $(S,\sigma)$ coordinates to illustrate the evolution of the sum of irreversible tensile and compressive strain $P^\ast(t,\mathbf{y},\mathbf{x},\mathbf{u})+P_\ast(t,\mathbf{y},\mathbf{x},\mathbf{u})$ At the beginning of this cycle $A$ there is no plastic strain, after tension loading there is tensile plastic stress $B$ after compressive loading the sum of the plastic stress under compression and tension is given by $C$. (b) A bond that degrades in tension and compression undergoing stress-strain cycling now plotted in $(E,\sigma)$ coordinates to show stiffness degradation under tension and compression. Here the cycle starts with strain loading and then unloads then reverses and is loaded in compression and again unloads. Both figures correspond to the same loading cycle with bilinear tensile loading followed by linear-exponential compression loading.
• Figure 5: Calibration of $\beta_\pm$ based on experimental data. $x_1$ is the $x$ axis intercept of a chosen unloading curve as measured by experiment and $x_2$ is the point of unloading that would occur without any loss in elasticity. The ratio of these values is the unloading ratio, $\beta_\pm = x_1/x_2$

Theorems & Definitions (7)

• Definition 1: Bond
• Definition 2
• Definition 3: Broken Bond
• Theorem 1: Existence and Uniqueness of Solution of the Displacement and Force Controlled Fracture Evolution
• Lemma 1
• Lemma 2: Growth of the dissipation for fracture in quasi-brittle materials
• Corollary 2.1: Condition for energy dissipation in fracture evolution for quasi brittle materials