Solid mechanics

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Solid mechanics (also known as mechanics of solids) is the branch of continuum mechanics that studies the behavior of solid materials, especially their motion and deformation under the action of forces, temperature changes, phase changes, and other external or internal agents.

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Solid mechanics is fundamental for civil, aerospace, nuclear, biomedical and mechanical engineering, for geology, and for many branches of physics and chemistry such as materials science. [1] It has specific applications in many other areas, such as understanding the anatomy of living beings, and the design of dental prostheses and surgical implants. One of the most common practical applications of solid mechanics is the Euler–Bernoulli beam equation. Solid mechanics extensively uses tensors to describe stresses, strains, and the relationship between them.

Solid mechanics is a vast subject because of the wide range of solid materials available, such as steel, wood, concrete, biological materials, textiles, geological materials, and plastics.

Fundamental aspects

A solid is a material that can support a substantial amount of shearing force over a given time scale during a natural or industrial process or action. This is what distinguishes solids from fluids, because fluids also support normal forces which are those forces that are directed perpendicular to the material plane across from which they act and normal stress is the normal force per unit area of that material plane. Shearing forces in contrast with normal forces, act parallel rather than perpendicular to the material plane and the shearing force per unit area is called shear stress.

Therefore, solid mechanics examines the shear stress, deformation and the failure of solid materials and structures.

The most common topics covered in solid mechanics include:

  1. stability of structures - examining whether structures can return to a given equilibrium after disturbance or partial/complete failure
  2. dynamical systems and chaos - dealing with mechanical systems highly sensitive to their given initial position
  3. thermomechanics - analyzing materials with models derived from principles of thermodynamics
  4. biomechanics - solid mechanics applied to biological materials e.g. bones, heart tissue
  5. geomechanics - solid mechanics applied to geological materials e.g. ice, soil, rock
  6. vibrations of solids and structures - examining vibration and wave propagation from vibrating particles and structures i.e. vital in mechanical, civil, mining, aeronautical, maritime/marine, aerospace engineering
  7. fracture and damage mechanics - dealing with crack-growth mechanics in solid materials
  8. composite materials - solid mechanics applied to materials made up of more than one compound e.g. reinforced plastics, reinforced concrete, fiber glass
  9. variational formulations and computational mechanics - numerical solutions to mathematical equations arising from various branches of solid mechanics e.g. finite element method (FEM)
  10. experimental mechanics - design and analysis of experimental methods to examine the behavior of solid materials and structures

Relationship to continuum mechanics

As shown in the following table, solid mechanics inhabits a central place within continuum mechanics. The field of rheology presents an overlap between solid and fluid mechanics.

Continuum mechanics
The study of the physics of continuous materials
Solid mechanics
The study of the physics of continuous materials with a defined rest shape.
Elasticity
Describes materials that return to their rest shape after applied stresses are removed.
Plasticity
Describes materials that permanently deform after a sufficient applied stress.
Rheology
The study of materials with both solid and fluid characteristics.
Fluid mechanics
The study of the physics of continuous materials which deform when subjected to a force.
Non-Newtonian fluid
Do not undergo strain rates proportional to the applied shear stress.
Newtonian fluids undergo strain rates proportional to the applied shear stress.

Response models

A material has a rest shape and its shape departs away from the rest shape due to stress. The amount of departure from rest shape is called deformation, the proportion of deformation to original size is called strain. If the applied stress is sufficiently low (or the imposed strain is small enough), almost all solid materials behave in such a way that the strain is directly proportional to the stress; the coefficient of the proportion is called the modulus of elasticity. This region of deformation is known as the linearly elastic region.

It is most common for analysts in solid mechanics to use linear material models, due to ease of computation. However, real materials often exhibit non-linear behavior. As new materials are used and old ones are pushed to their limits, non-linear material models are becoming more common.

These are basic models that describe how a solid responds to an applied stress:

  1. Elasticity – When an applied stress is removed, the material returns to its undeformed state. Linearly elastic materials, those that deform proportionally to the applied load, can be described by the linear elasticity equations such as Hooke's law.
  2. Viscoelasticity – These are materials that behave elastically, but also have damping: when the stress is applied and removed, work has to be done against the damping effects and is converted in heat within the material resulting in a hysteresis loop in the stress–strain curve. This implies that the material response has time-dependence.
  3. Plasticity – Materials that behave elastically generally do so when the applied stress is less than a yield value. When the stress is greater than the yield stress, the material behaves plastically and does not return to its previous state. That is, deformation that occurs after yield is permanent.
  4. Viscoplasticity - Combines theories of viscoelasticity and plasticity and applies to materials like gels and mud.
  5. Thermoelasticity - There is coupling of mechanical with thermal responses. In general, thermoelasticity is concerned with elastic solids under conditions that are neither isothermal nor adiabatic. The simplest theory involves the Fourier's law of heat conduction, as opposed to advanced theories with physically more realistic models.

Timeline

Galileo Galilei published the book "Two New Sciences" in which he examined the failure of simple structures Galileo Galilei by Ottavio Leoni Marucelliana (cropped).jpg
Galileo Galilei published the book "Two New Sciences" in which he examined the failure of simple structures
Isaac Newton published "Philosophiae Naturalis Principia Mathematica" which contains the Newton's laws of motion Sir Isaac Newton by Sir Godfrey Kneller, Bt.jpg
Isaac Newton published "Philosophiae Naturalis Principia Mathematica" which contains the Newton's laws of motion
Leonhard Euler developed the theory of buckling of columns Leonhard Euler 2.jpg
Leonhard Euler developed the theory of buckling of columns

See also

Related Research Articles

Rheology is the study of the flow of matter, primarily in a fluid state, but also as "soft solids" or solids under conditions in which they respond with plastic flow rather than deforming elastically in response to an applied force. Rheology is a branch of physics, and it is the science that deals with the deformation and flow of materials, both solids and liquids.

<span class="mw-page-title-main">Stress (mechanics)</span> Physical quantity that expresses internal forces in a continuous material

In continuum mechanics, stress is a physical quantity that describes forces present during deformation. An object being pulled apart, such as a stretched elastic band, is subject to tensile stress and may undergo elongation. An object being pushed together, such as a crumpled sponge, is subject to compressive stress and may undergo shortening. The greater the force and the smaller the cross-sectional area of the body on which it acts, the greater the stress. Stress has units of force per area, such as newtons per square meter (N/m2) or pascal (Pa).

<span class="mw-page-title-main">Plasticity (physics)</span> Non-reversible deformation of a solid material in response to applied forces

In physics and materials science, plasticity is the ability of a solid material to undergo permanent deformation, a non-reversible change of shape in response to applied forces. For example, a solid piece of metal being bent or pounded into a new shape displays plasticity as permanent changes occur within the material itself. In engineering, the transition from elastic behavior to plastic behavior is known as yielding.

In physics and materials science, elasticity is the ability of a body to resist a distorting influence and to return to its original size and shape when that influence or force is removed. Solid objects will deform when adequate loads are applied to them; if the material is elastic, the object will return to its initial shape and size after removal. This is in contrast to plasticity, in which the object fails to do so and instead remains in its deformed state.

Structural analysis is a branch of solid mechanics which uses simplified models for solids like bars, beams and shells for engineering decision making. Its main objective is to determine the effect of loads on the physical structures and their components. In contrast to theory of elasticity, the models used in structure analysis are often differential equations in one spatial variable. Structures subject to this type of analysis include all that must withstand loads, such as buildings, bridges, aircraft and ships. Structural analysis uses ideas from applied mechanics, materials science and applied mathematics to compute a structure's deformations, internal forces, stresses, support reactions, velocity, accelerations, and stability. The results of the analysis are used to verify a structure's fitness for use, often precluding physical tests. Structural analysis is thus a key part of the engineering design of structures.

An elastic modulus is the unit of measurement of an object's or substance's resistance to being deformed elastically when a stress is applied to it.

Stress–strain analysis is an engineering discipline that uses many methods to determine the stresses and strains in materials and structures subjected to forces. In continuum mechanics, stress is a physical quantity that expresses the internal forces that neighboring particles of a continuous material exert on each other, while strain is the measure of the deformation of the material.

<span class="mw-page-title-main">Beam (structure)</span> Structural element capable of withstanding loads by resisting bending

A beam is a structural element that primarily resists loads applied laterally to the beam's axis. Its mode of deflection is primarily by bending. The loads applied to the beam result in reaction forces at the beam's support points. The total effect of all the forces acting on the beam is to produce shear forces and bending moments within the beams, that in turn induce internal stresses, strains and deflections of the beam. Beams are characterized by their manner of support, profile, equilibrium conditions, length, and their material.

In materials science and continuum mechanics, viscoelasticity is the property of materials that exhibit both viscous and elastic characteristics when undergoing deformation. Viscous materials, like water, resist shear flow and strain linearly with time when a stress is applied. Elastic materials strain when stretched and immediately return to their original state once the stress is removed.

<span class="mw-page-title-main">Fracture mechanics</span> Study of propagation of cracks in materials

Fracture mechanics is the field of mechanics concerned with the study of the propagation of cracks in materials. It uses methods of analytical solid mechanics to calculate the driving force on a crack and those of experimental solid mechanics to characterize the material's resistance to fracture.

In physics and engineering, a constitutive equation or constitutive relation is a relation between two physical quantities that is specific to a material or substance, and approximates the response of that material to external stimuli, usually as applied fields or forces. They are combined with other equations governing physical laws to solve physical problems; for example in fluid mechanics the flow of a fluid in a pipe, in solid state physics the response of a crystal to an electric field, or in structural analysis, the connection between applied stresses or loads to strains or deformations.

<span class="mw-page-title-main">Bending</span> Strain caused by an external load

In applied mechanics, bending characterizes the behavior of a slender structural element subjected to an external load applied perpendicularly to a longitudinal axis of the element.

This is an alphabetical list of articles pertaining specifically to structural engineering. For a broad overview of engineering, please see List of engineering topics. For biographies please see List of engineers.

This is an alphabetical list of articles pertaining specifically to Engineering Science and Mechanics (ESM). For a broad overview of engineering, please see Engineering. For biographies please see List of engineers and Mechanicians.

In continuum mechanics, Lamé parameters are two material-dependent quantities denoted by λ and μ that arise in strain-stress relationships. In general, λ and μ are individually referred to as Lamé's first parameter and Lamé's second parameter, respectively. Other names are sometimes employed for one or both parameters, depending on context. For example, the parameter μ is referred to in fluid dynamics as the dynamic viscosity of a fluid(not the same units); whereas in the context of elasticity, μ is called the shear modulus, and is sometimes denoted by G instead of μ. Typically the notation G is seen paired with the use of Young's modulus E, and the notation μ is paired with the use of λ.

Material failure theory is an interdisciplinary field of materials science and solid mechanics which attempts to predict the conditions under which solid materials fail under the action of external loads. The failure of a material is usually classified into brittle failure (fracture) or ductile failure (yield). Depending on the conditions most materials can fail in a brittle or ductile manner or both. However, for most practical situations, a material may be classified as either brittle or ductile.

<span class="mw-page-title-main">Deformation (physics)</span> Transformation of a body from a reference configuration to a current configuration

In physics and continuum mechanics, deformation is the transformation of a body from a reference configuration to a current configuration. A configuration is a set containing the positions of all particles of the body.

<span class="mw-page-title-main">Structural engineering theory</span>

Structural engineering depends upon a detailed knowledge of loads, physics and materials to understand and predict how structures support and resist self-weight and imposed loads. To apply the knowledge successfully structural engineers will need a detailed knowledge of mathematics and of relevant empirical and theoretical design codes. They will also need to know about the corrosion resistance of the materials and structures, especially when those structures are exposed to the external environment.

<span class="mw-page-title-main">Flow plasticity theory</span>

Flow plasticity is a solid mechanics theory that is used to describe the plastic behavior of materials. Flow plasticity theories are characterized by the assumption that a flow rule exists that can be used to determine the amount of plastic deformation in the material.

Johannes Ferdinand "Hans" Besseling was professor emeritus of Engineering Mechanics at the Delft University of Technology, worked in the field of the application of solid mechanics to the analysis of structures; constitutive equations for the mathematical description of material behaviour. His specialities are finite element methods, continuum thermodynamics, creep and plasticity of metals.

References

Notes

  1. Allan Bower (2009). Applied mechanics of solids. CRC press. Retrieved March 5, 2017.

Bibliography