# Deformation (mechanics)

**straindeformationshear strainstrainselongationmechanical straindeformationsdeformedFinite deformation tensorsdeform**

Deformation in continuum mechanics is the transformation of a body from a reference configuration to a current configuration.wikipedia

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### Constitutive equation

**constitutive relationconstitutive equationsconstitutive model**

The relation between stresses and induced strains is expressed by constitutive equations, e.g., Hooke's law for linear elastic materials.

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 forces to strains or deformations.

### Linear elasticity

**elastic waveselastic wavelinear elastic**

The relation between stresses and induced strains is expressed by constitutive equations, e.g., Hooke's law for linear elastic materials.

The fundamental "linearizing" assumptions of linear elasticity are: infinitesimal strains or "small" deformations (or strains) and linear relationships between the components of stress and strain.

### Hooke's law

**spring constantforce constantelasticity tensor**

The relation between stresses and induced strains is expressed by constitutive equations, e.g., Hooke's law for linear elastic materials.

The modern theory of elasticity generalizes Hooke's law to say that the strain (deformation) of an elastic object or material is proportional to the stress applied to it.

### Viscoelasticity

**viscoelasticvisco-elasticVisco-elasticity**

Another type of irreversible deformation is viscous deformation, which is the irreversible part of viscoelastic deformation.

Viscous materials, like water, resist shear flow and strain linearly with time when a stress is applied.

### Yield (engineering)

**yield strengthyield stressyield**

One type of irreversible deformation is plastic deformation, which occurs in material bodies after stresses have attained a certain threshold value known as the elastic limit or yield stress, and are the result of slip, or dislocation mechanisms at the atomic level.

Yield occurs when the maximum principal strain reaches the strain corresponding to the yield point during a simple tensile test.

### Fluid

**fluidsanalysis of fluidsenergy fluids**

In physics, a fluid is a substance that continually deforms (flows) under an applied shear stress, or external force.

### Elastomer

**elasticelastomerselastomeric**

On the other hand, for some materials, e.g. elastomers and polymers, subjected to large deformations, the engineering definition of strain is not applicable, e.g. typical engineering strains greater than 1%, thus other more complex definitions of strain are required, such as stretch, logarithmic strain, Green strain, and Almansi strain.

An elastomer is a polymer with viscoelasticity (i.e., both viscosity and elasticity) and has very weak intermolecular forces, generally low Young's modulus and high failure strain compared with other materials.

### Infinitesimal strain theory

**strain tensorstraininfinitesimal strain tensor**

In continuum mechanics, the infinitesimal strain theory is a mathematical approach to the description of the deformation of a solid body in which the displacements of the material particles are assumed to be much smaller (indeed, infinitesimally smaller) than any relevant dimension of the body; so that its geometry and the constitutive properties of the material (such as density and stiffness) at each point of space can be assumed to be unchanged by the deformation.

### Deformation (engineering)

**deformationplastic deformationelastic deformation**

Deformation is often described as strain.

### Shear stress

**shearshearingwall shear stress**

These definitions are consistent with those of normal stress and shear stress.

Pure shear stress is related to pure shear strain, denoted γ, by the following equation:

### Continuum mechanics

**continuumcontinuous mediacontinuum physics**

Deformation in continuum mechanics is the transformation of a body from a reference configuration to a current configuration.

The internal contact forces are related to the body's deformation through constitutive equations.

### Soft tissue

**soft tissuessoft-tissueanimal tissue**

At small strains, elastin confers stiffness to the tissue and stores most of the strain energy.

### Force

**forcesattractiveelastic force**

In a continuous body, a deformation field results from a stress field induced by applied forces or is due to changes in the temperature field inside the body.

The stress tensor accounts for forces that cause all strains (deformations) including also tensile stresses and compressions.

### Tensor

**tensorsorderclassical treatment of tensors**

A strain is in general a tensor quantity.

### Dimensionless quantity

**dimensionlessdimensionless numberdimensionless quantities**

A more complex example of such a ratio is engineering strain, a measure of physical deformation defined as a change in length divided by the initial length.

### Parts-per notation

**ppmparts per millionppb**

Hence strains are dimensionless and are usually expressed as a decimal fraction, a percentage or in parts-per notation.

### Plasticity (physics)

**plasticityplasticplastic deformation**

However, even ductile metals will fracture when the strain becomes large enough—this is as a result of work hardening of the material, which causes it to become brittle.

### Simple shear

**shearplane shearshearing**

A simple shear deformation is defined as an isochoric plane deformation in which there is a set of line elements with a given reference orientation that do not change length and orientation during the deformation.

Simple shear is a deformation in which parallel planes in a material remain parallel and maintain a constant distance, while translating relative to each other.

### Displacement field (mechanics)

**displacementdisplacement fielddisplacements**

The deformation is described by the displacement field

For example, a displacement field may be used to describe the effects of deformation on a solid body.

### Finite strain theory

**deformation gradientfinite strainfinite deformations**

, the deformation gradient has the form

In continuum mechanics, the finite strain theory—also called large strain theory, or large deformation theory—deals with deformations in which strains and/or rotations are large enough to invalidate assumptions inherent in infinitesimal strain theory.

### Stress (mechanics)

**stressstressestensile stress**

These definitions are consistent with those of normal stress and shear stress. In a continuous body, a deformation field results from a stress field induced by applied forces or is due to changes in the temperature field inside the body.

In continuum mechanics, stress is a physical quantity that expresses the internal forces that neighbouring particles of a continuous material exert on each other, while strain is the measure of the deformation of the material, which is not a physical quantity.

### Shear modulus

**modulus of rigidityshear moduliRigidity modulus**

In materials science, shear modulus or modulus of rigidity, denoted by G, or sometimes S or μ, is defined as the ratio of shear stress to the shear strain:

### Pure shear

It is an example of irrotational strain in which body is elongated in one direction while being shortened perpendicularly.

### Moiré pattern

**moirémoiremoire pattern**

In manufacturing industries, these patterns are used for studying microscopic strain in materials: by deforming a grid with respect to a reference grid and measuring the moiré pattern, the stress levels and patterns can be deduced.