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*Definition:* A material that behaves as a solid until a certain threshold of stress, known as the critical stress or yield stress, is exceeded. Once this threshold is surpassed, the material begins to flow like a liquid.

**Characteristics:**

**Critical Stress:**

**Yield Stress:**Bingham solids do not exhibit flow behavior until the applied stress exceeds a specific value. This stress is known as the yield stress or critical stress. Below this threshold, the material remains rigid or behaves like a solid.

**Flow Behavior:**

**Post-Yield:**Once the critical stress is exceeded, the material transitions to a more fluid-like behavior, allowing it to flow or deform. The flow behavior typically follows a linear relationship with the applied stress above the yield stress.

**Examples:**

**Toothpaste:**Remains in a solid-like state until squeezed, at which point it flows out of the tube.**Modeling Clay:**Maintains its shape until sufficient force is applied to deform or shape it.

**Mathematical Model:**

**Bingham Plastic Model:**The flow behavior of Bingham solids is often described by the Bingham plastic model. The relationship between shear stress\((\tau)\) and shear rate\((\dot{\gamma})\) is given by:\[

\tau = \tau_y + \eta_p \dot{\gamma}

\]

where:\(\tau_y\) is the yield stress,\(\eta_p\) is the plastic viscosity,\(\dot{\gamma}\) is the shear rate.

\((\tau < \tau_y)\) , the material does not flow. Above this threshold, the material exhibits a linear relationship between shear stress and shear rate, indicating its flow behavior.

**Applications and Implications:**

**Industrial Use:**

**Manufacturing:**Understanding the Bingham behavior of materials is crucial in industries where precise flow characteristics are required, such as in food processing, pharmaceuticals, and materials handling.**Product Design:**For products like toothpaste or paints, designing containers and application mechanisms takes into account the yield stress to ensure proper dispensing and performance.

**Rheology:**

**Study of Flow:**The study of Bingham solids falls under rheology, the science of flow and deformation of materials. Accurate modeling of Bingham solids helps in predicting and controlling the behavior of these materials in various applications.

**Summary:**

A Bingham solid is a material that behaves like a solid until a critical stress (yield stress) is applied. Once this stress is exceeded, the material begins to flow like a liquid. This behavior is described by the Bingham plastic model and is important in various industrial and manufacturing applications.

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