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Continuum mechanics is the branch of mechanics that describes solids, liquids and gases as continuous matter rather than as collections of individual atoms. It expresses what happens to a body under load or flow through a fixed set of balance laws (mass, momentum, energy) plus a material-specific constitutive equation, and it is the theory underneath structural, fluid and thermomechanical simulation. This page is for researchers and lab staff who run tests or finite element models and want to know what the theory does and does not say.
Short answer
Continuum mechanics models a material body as a continuous medium in which quantities such as density, stress and temperature are defined at every point. The balance laws are the same for every material; the constitutive equation is what distinguishes steel from water from rubber, and it is the part that laboratory measurements supply.
Definition and scope
In Z. Martinec’s lecture notes Continuum Mechanics for Geophysicists and Geodesists, Part I (Universität Stuttgart, Report 1999.2), the subject considers “material bodies in the form of solids, liquids, and gases”, and a continuum particle is essentially a material point for which a density is defined, as opposed to a classical-mechanics particle that is assigned a mass. K. Garikipati’s open Lectures on Continuum Physics (University of Michigan, 2013) takes the same scope and places equal emphasis on solids and fluids, arguing that one cannot fully appreciate the mechanical state of one without the other.
Treating matter as continuous is a modelling choice, not a claim that matter has no atoms. It is appropriate when the length scale of interest is much larger than the spacing of the atomic structure, so that point values can be read as averages over small material volumes. At very small scales the choice has to be re-examined; see the nanoscale guide for how the range is defined in lab practice.
How it is used in practice: the modelling chain
Both course sources build the subject in the same order. That order doubles as a checklist for setting up or reviewing a mechanics model.
- Kinematics (strain and deformation). Describe how the body moves and deforms. Martinec separates a reference configuration (for example the initial one) from the current configuration, and material (Lagrangian) coordinates from spatial (Eulerian) coordinates. Garikipati’s units cover the deformation gradient and polar decomposition.
- Stress. Define body forces and surface forces. Martinec introduces the Cauchy traction principle: the limit of the resultant surface force over an area, as the area shrinks to zero at an interior point, exists and defines the traction.
- Balance laws. Martinec lists the local forms as the continuity equation, the equation of motion, the symmetry of the Cauchy stress tensor, the energy equation and the entropy inequality. Garikipati covers balance of mass, linear and angular momentum, the Reynolds transport theorem and energy balance.
- Constitutive equations. Martinec treats the need for constitutive equations, then elastic materials, incompressible materials, linear elasticity and Hooke’s law for isotropic media. Garikipati adds objectivity and material symmetry.
- Boundary and initial value problems. Garikipati includes a summary of initial and boundary value problems of continuum mechanics; this is the step that a simulation package actually solves.
The practical consequence: steps 1 to 3 hold for any material, so the unknowns outnumber the equations until step 4 supplies a material law. Parameters in that law (for example elastic constants in Hooke’s law) are what mechanical tests estimate. The solid mechanics guide covers the test standards and verification and validation practice that feed that step, and the LS-DYNA guide shows one solver where these ingredients appear as element formulations and material cards.
Branches and related concepts
The table follows the unit structure of Garikipati’s lectures and the scope stated by Martinec; each row names the source that places the topic inside continuum theory.
| Branch | What it adds | Where the source covers it |
|---|---|---|
| Solid mechanics | Deformation, stress and elastic response of solids | Garikipati units on elasticity; Martinec chapter 5 (elastic materials, Hooke’s law) |
| Fluid mechanics | Same balance laws, with a constitutive law for fluids and incompressibility | Garikipati (equal solid and fluid emphasis); Martinec (incompressible materials) |
| Thermomechanics | Coupling of mechanics with energy balance and the entropy inequality | Garikipati unit 11; Martinec chapter 4 (energy equation, entropy inequality) |
| Variational principles | Energy-based formulations of the governing equations, related to calculus of variations | Garikipati unit 12 |
| Mass transport | Diffusion-type processes such as Cahn-Hilliard, treated as continuum physics | Garikipati unit 13 |
Where it matters for labs and materials work
- Interpreting test data. A tensile or compression test reports force and displacement; continuum mechanics is the framework that converts these into stress, strain and a constitutive parameter. Which definition of strain and which configuration you use changes the numbers at large deformation.
- Simulation set-up. Finite element and related solvers discretise the balance laws; the material model you choose is the constitutive equation. See the materials science guide for how that connects to characterisation.
- Reading the literature. Papers state results in either Lagrangian or Eulerian description. Knowing which is in use is needed to compare quantities between studies.
Common misunderstandings
Does continuum mechanics assume matter has no atoms?
No. It is a model that is valid where the scale of interest is much larger than the atomic scale. It makes no statement about atomic structure itself.
Do the balance laws describe a particular material?
No. Mass, momentum and energy balance apply to every continuum body. A material is specified only by its constitutive equation, which must also be consistent with the entropy inequality (Martinec, chapter 4).
Is Eulerian or Lagrangian the correct description?
Neither is more correct. They are two coordinate descriptions of the same motion: material coordinates follow a particle, spatial coordinates fix a point in space (Martinec, chapter 2). Solvers and textbooks choose by convenience.
Is it only about solids?
No. Both sources cover solids and fluids; Martinec’s notes name solids, liquids and gases explicitly.
FAQ
What is continuum mechanics in one sentence?
It is the mathematical description of the deformation and flow of solids, liquids and gases, treating them as continuous media governed by balance laws and material-specific constitutive equations.
What maths does it need?
Martinec’s notes are written for a one-semester course for undergraduates with a year of calculus. Garikipati’s lectures begin with vectors and tensors and state plainly that the treatment is mathematical.
How is continuum mechanics different from continuum physics?
Garikipati uses “continuum physics” for the wider course: the first ten units are standard continuum mechanics, and the last three (thermomechanics, variational principles, mass transport) go beyond it.
Sources
- Martinec, Z. (1999). Continuum Mechanics for Geophysicists and Geodesists, Part I: Basic Theory. Report Nr. 1999.2, Universität Stuttgart. PDF. Read 2026-11-11.
- Garikipati, K. Lectures on Continuum Physics. Open Michigan, University of Michigan (2013, revised 2014; CC BY-NC 3.0). Course page. Read 2026-11-11.
