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When analyzing one-dimensional motion with constant acceleration, the problem-solving strategy involves identifying the known quantities and choosing the appropriate kinematic equations to solve for the unknowns. Either one or two kinematic equations are needed to solve for the unknowns, depending on the known and unknown quantities. Generally, the number of equations required is the same as the number of unknown quantities in the given example. Two-body pursuit problems always require two...
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The second kinematic equation expresses the final position of an object in terms of its initial position, the distance traveled with the initial constant velocity, and the distance traveled due to a change in velocity. Similar to the first kinematic equation, this equation is also only valid when the acceleration is constant throughout the motion of an object.
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The first two kinematic equations have time as a variable, but the third kinematic equation is independent of time. This equation expresses final velocity as a function of the acceleration and distance over which it acts. The fourth kinematic equation does not have an acceleration term and provides the final position of the object at time t in terms of the initial and final velocities. This equation is useful when the value of the constant acceleration is unknown.
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When an object moves with constant acceleration, the velocity of the object changes at a constant rate throughout the motion. The kinematic equations of motions are derived for such cases where the acceleration of the object is constant. The first kinematic equation gives an insight into the relationship between velocity, acceleration, and time. We can see, for example:
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In mechanics, when one observes a rigid body in rotational motion with constant angular acceleration, it is possible to establish equations for its rotational kinematics. This process resembles how linear kinematics are dealt with in simpler motion studies.
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Kinematics is the description of motion. The kinematics of rotational motion discusses the relationships between rotation angle, angular velocity, angular acceleration, and time. One can describe many things with great precision using kinematics, but kinematics does not consider causes. For example, a large angular acceleration describes a very rapid change in angular velocity without any consideration of its cause. Thus, rotational kinematics does not represent the laws of nature.
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Continuum kinematics with incompatible-compatible decomposition.

Vladimir Goldshtein1, Paolo Maria Mariano2, Domenico Mucci3

  • 1Department of Mathematics, Ben-Gurion University of the Negev, Israel.

Philosophical Transactions. Series A, Mathematical, Physical, and Engineering Sciences
|November 5, 2023
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Summary
This summary is machine-generated.

This study introduces a new framework for anelastic deformation, detailing how material structure changes during processes like plasticity and fracture. It offers a clear decomposition method applicable to continuum mechanics problems.

Keywords:
compatibilitycontinuum mechanicselastic–plastic decompositionincompatibilitykinematicsvector bundle morphisms

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Area of Science:

  • Continuum Mechanics
  • Material Science
  • Geometric Mechanics

Background:

  • Anelastic deformation involves changes in material structure and geometric configuration.
  • Existing models may not fully capture complex phenomena like plasticity and fracture.

Purpose of the Study:

  • To present a novel framework for the kinematics of anelastic deformation.
  • To develop an unambiguous decomposition for analyzing material behavior.

Main Methods:

  • Developing a kinematic framework for material bodies.
  • Constructing an unambiguous decomposition into incompatible and compatible factors.

Main Results:

  • The proposed framework accommodates changes in material and geometric structure.
  • It includes a decomposition that encompasses standard elastic-plastic decomposition.
  • The framework is relevant to plasticity, fracture, and non-injective deformations.

Conclusions:

  • The new framework provides a robust method for analyzing anelastic deformation.
  • It offers a unified approach to understanding complex material behaviors in continuum mechanics.