At room temperature, play-dough is solid(ish). But if you make a thin strip it cannot just stand up on it's own, so is it still solid?
On a more general note, what classifies or differentiates a solid from a liquid?
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Play-Doh is mostly flour, salt and water, so it's basically just (unleavened) dough. There are a lot of extra components like colourings, fragrances, preservatives etc, but these are present at low levels and don't have a huge effect on the rheology.
The trouble with saying it's basically just dough is that the rheology of dough is fearsomely complicated. In a simple flour/salt/water dough you have a liquid phase made up of an aqueous solution of polymers like gluten, and solid particles of starch. So a dough is basically a suspension of solid particles in a viscous fluid. To make things more complicated the particles are flocculated, so you end up with a material that exhibits a yield stress unlike the non-flocculated particles in e.g. oobleck.
At low stresses dough behaves like a solid because the flocculated particles act like a skeleton. However the bonds between flocculated particles are weak (they're only Van der Waals forces) so at even moderate stresses the dough flows and behaves like a liquid. Dough, and Play-Doh, are best described as non-Newtonian fluids.
I'm going to guess the toy you actually have in mind is the stuff sold in the US under the name Silly Putty . Play-Doh is used primarily as a "modeling clay" for sculpture - which means it needs to behave as a "plastic" - it's yield strength needs be low enough to enable it to be worked into a figure, but high enough that reasonable sized figures don't quickly collapse.
Silly Putty is quite dramatic (for putty) and often used as a teaching-aid to demonstrate a rheid - a now somewhat obsolete term for a material that exhibits a wide range of mechanical behavior depending upon the magnitude of stress. Silly Putty fractures if hit with a hammer, bounces like rubber, stretches like metal, and can flow through a hole like a fluid.
A constitutive law is a phenomenological law that describes the relation of load to the strain and strain-rate of a material. Physicists, chemists, and engineers sometimes adopt a linear viscoelastic constitutive law to model the deformation of a material. The first constitutive equation (constitutive law) was developed by Robert Hooke (Hooke's law.) Viscoelastic constitutive laws can be useful because they are relatively easy to handle mathematically and because they have simple intuitive analogs: a spring (elastic material,) a dashpot (newton viscous material) and a sliding block with friction (plastic material.) You can also build more complex constitutive laws by combining these simple analogs in series or in parallel. For example the constitutive law for Play-doh (plastic) might be modeled by a block sliding on a surface with friction - whenever the friction force is exceeded the analog block slides (but then does not move back when the load is removed i.e. no spring.) Silly putty might be better modeled by a block – spring - dashpot connected in series. A sudden tension can move the spring only (elastic behavior) a larger tension can move the block (plastic) and a tension applied for a long time could move the dashpot. There are also materials that are observed to have non-linear constitutive behavior that is not well described by any linear viscoelastic model.
The words “liquid” and “solid” can also refer to state-of-matter of a phase in a thermodynamic system. Phases are well defined in a thermodynamic system because they are usually separated by phase boundaries and energy transitions. State-of-matter and constitutive phenomenology are not identical.
You should probably not classify a material as a solid or a liquid based upon its constitutive behavior. Those words are better used to refer to different phases of a thermodynamic system.
You should not refer to the state of matter when you mean constitutive behavior – they are different things. Water filling a lake is in the liquid state, but also behaves as an elastic material. Ice in a glacier is in the solid state but flows downhill.
When trying to understand the rheology of materials, it is important to consider the influence of time.
Geophysicists realized in the 20th the century that the rocks of the Earth's crust and mantle also behave as a 'rheid' - but over geological time scales. This fact had to be understood by earth scientists in order to explain/model the underlying physics of modern planetary tectonophysics, and the observations of continental drift etc..
While the physical properties of a solid vs a liquid are obvious to any grade-schooler, the physics behind it are a little more complex.
A substance is traditionally called a solid if it will not noticeably deform from a given starting shape in its steady state (in simple terms, it will not "flow" in the absence of any force other than gravity). A liquid, on the other hand, will. Typically, a homogeneous solid material is solid because the molecules have insufficient thermal energy to overcome intermolecular attractions (such as hydrogen bonding, Van der Waals forces, etc) that give it its rigidity. This energy can come from many places; classically it's thought of as thermal energy (heat) so that we can describe things in terms of ambient temperature and pressure. However, under different levels of temperature and pressure that would be "normal" in situations other than at sea level on Earth, we get other forms and states of matter beyond the three traditional ones taught in grade school. As the commenter mentioned, ice is a crystalline solid, but given enough mass of it to exert pressure, such as in glaciers, it will still "flow"; likewise, metals are traditionally solid, but their malleability allows them to be shaped under sufficient force, including the pull of gravity on its own weight.
In between classical solids and liquids, we typically have a few substances that have some properties of both. The overwhelming majority are chemically-heterogeneous, but microscopically-homogeneous mixtures of a solid and a liquid, which are called "colloids" or "colloidal suspensions". If the mixture is more solid than liquid, and will not noticeably "flow" under the force of gravity alone, we typically call it a "gel". If the substance is more liquid than solid, and behaves as such (as a thick liquid), it is a "sol". Play-Doh falls pretty far into the "gel" category; it is highly "plastic" (flexible, moldable, deformable) but in most situations it won't deform under gravity alone. The exceptions come when the material is in a shape such that there isn't enough of the material for the intermolecular forces to withstand the force of gravity (such as when a thin shape of the material is extruded).
John's suggestion of a "non-Newtonian fluid" is clever, but the physics of the material say otherwise. A fluid is non-Newtonian, simply, if its rate of deformation is not directly proportional to the magnitude of the forces acting on it. Oobleck (corn starch & water) is such a material because it becomes thicker and even solidifies under high stresses such as trying to squeeze it, but will readily flow under lower stresses such as gravity. Play-Doh's behavior is much more proportional; it will deform when squeezed and molded but stay put, more or less, under gravity alone.